Introduction to Uniform Spaces

Lxjndon Mathematical Society Lecture Note Series. 144

Introduction to Uniform Spaces I.M. James Savilian Professor of Gcomclry Malhemalical Insiiiuie, Uiiiversily of Oxford

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C A M B R n X i E UNIVERSITY PRESS Cambridge New York

Port Chester Melbourne

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Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011, USA 10, Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1990 First published 1990 Printed in Great Britain at the University Press, Cambridge Library of Congress cataloguing in publication data available British Library cataloguing in publication data available

ISBN 0 521 38620 9

(.1

Contents

Introduction 1.

Uniform

structures

2.

I n d u c e d and c o i n d u c e d u n i f o r m

3.

The u n i f o r m

4.

Completeness

5.

Topological

6.

Uniform

transformation

7.

Uniform

s p a c e s over a base

8.

Uniform c o v e r i n g Append i x :

topology completion

groups

spaces

f iIter s

Exercises Bibliography Index

and

groups

structures

iniroauciion

T h i s b o o k i s b a s e d on a c o u r s e given at

the University

and g r a d u a t e

students.

but t h e remainder i s published

of Oxford

Here f o r

and o t h e r s

in

subject

in Chapter

at l e a s t

an o u t l i n e

II

of

Bourbaki

text

several

s p e c i a l i z e d monographs,

Roelcke

and D i e r o l f

to general

many f i n d

chapters

topological

stage.

no k n o w l e d g e o f

topology.

b a s i c k n o w l e d g e of

topology

u n i f o r m w o r l d and t h e

Γ 2 ].

it

is

[25]

the

However,

on t o p o l o g y

such as I s b e l l

the

contain are

[ 7 J, P a g e

[19J,

others. the transition is

from

metric

a major s t e p

Uniform

I have t h e r e f o r e

i n s u c h a way t h a t

a c c o u n t of

Moreover t h e r e

spaces

extremely d i f f i c u l t .

intermediate

books

amongst

For s t u d e n t s o f m a t h e m a t i c s spaces

classical

I n d e e d much o f

The c l a s s i c

the theory.

[21],

undergraduati

s p a c e s was d e v e l o p e d by W e i l

t h e more r e c e n t of

recently

time.

the t h i r t i e s .

of

lectures

the m a t e r i a l i s

not so well-known.

the f i r s t

majority

sixteen

t o an a u d i e n c e o f

About h a l f

The t h e o r y o f u n i f o r m

is

of

which

s p a c e s make an

w r i t t e n the

excellent

first

two

t h e y c a n b e r e a d by a s t u d e n t The s e c o n d two c h a p t e r s

with

assume a

and a r e aimed a t s h o w i n g how t h e

t o p o l o g i c a l world are

related.

The m o n o g r a p h s on t h e t h e o r y m e n t i o n e d a b o v e are w r i t t e n tr.ainly

w i t h t h e n e e d s of

analysts

in mind.

Rather

o v e r t h e same g r o u n d a g a i n I h a v e c h o s e n t o e x p l o r e aspect

of

the s t a r t

the theory. that

Although

it

t h a n go

a different

has been r e c o g n i z e d

tODoloaical q r o u p s c a n p r o f i t a b l y

be

from

regarded

as unLXorm s p a c e s , that

it

is

I do n o t b e l i e v e

possible

groups.

t o d e v e l o p a t h e o r y of uniform

on n a t u r a l l y

to the

in the f i n a l

presented

t h e o r y of u n i f o r m

a b a s e and h e n c e t o t h e t h e o r y o f

uniform

transformation here.

spaces

covering

over

forty

find

exercises,

a l s o an a p p e n d i x c o n t a i n i n g

a short bibliography

t2J.

an o u t l i n e

book I h a v e ,

[21 J.

Ling for

[ 2 4 J.

and P r o f e s s o r some h e l p f u l

final

draft

e s c a p e d my

of

Finally,

the r e s u l t s

the

concerning

greatly relied

of Arens [ I J , like

on

by t h e work

of

consulted

Collins

t o t h a n k Mr.

Don S h i m a m o t o , w h o a t t e n d e d t h e

suggestions;

is

text.

of c o u r s e ,

I would

literature

There

Of t h e many p a p e r s I h a v e

I would p a r t i c u l a r l y m e n t i o n t h o s e and d e V r i e s

of

A l s o I h a v e b e e n much i n f l u e n c e d

R o e l c k e and D i e r o l f

and

mainly derived xron t h e

w h i c h a r e n e e d e d i n t h e main

In w r i t i n g t h i s Bourbaki

over

spaces,

on t h e s u b j e c t b u t w i t h some new p r o b l e m s a s w e l l .

filters

appreciated

section.

At t h e e n d t h e r e a d e r w i l l of

has been l u i i y

An o u t l i n e o f s u c h a t h e o r y i s

This leads

a set

it

Γ4] Paul

lectures,

l a t t e r very kindly

read

t h e t e x t and d e t e c t e d v a r i o u s e r r o r s w h i c h

the had

attention.

Mathematical I n s t i t u t e , J u l y 1989

Oxford

1.

Uniform structures

Let u s s t a r t

by r e c a l l i n g

from

t h e t h e o r y of r e l a t i o n s .

set

X

is

We w r i t e to

just ζΗη

η.

a subset when

Given

ξ

R

(ζ,η)

some n o t a t i o n

and

terminology

Formally a r e l a t i o n of

the c a r t e s i a n

square

and s a y t h a t

ξ

is

R-related

For e a c h s u b s e t

Η

of

€ R

we d e n o t e

on a g i v e n X χ X.

by

R[5] = {η : ζΚη} the s e t

of

R-relatives

of

ζ.

X

we

write R[H] = Note t h a t

υ RCei CcH

if

.

{Hj} i s

a family

of

s u b s e t s of

X

then

R[uHj ] = u R [ H j ] ; in g e n e r a l ,

however,

R[nH^ ]

The i d e n t i t y r e l a t i o n X X X.

A relation

reflexiv e. given

The r e v e r s e

on

a proper

X

is

nR[H^J.

j u s t the diagonal

which c o n t a i n s of

s u b s e t of

ΔΧ

a relation

R

is is

said

to

ΔΧ

of

be

the r e l a t i o n

R^

by if

We d e s c r i b e

R

and o n l y i f as symmetric

identity relation

is

The c o n p o s i t i o n is

R

is

the r e l a t i o n

ηΚξ

.

if

R = R ^;

for example

the

symmetric. of r e l a t i o n s

R S

on

X

R,

g i v e n by

S

on t h e ζ(Η

same s e t

» S)n

if

X

and onl'i

i-

I.M. James

if

ς5ζ

and

Also note

ζΗη

associative

ξ

Composition of

(but g e n e r a l l y

not commutative)

is

Η

for repeated compositions

length

such t h a t

i = 0 , . . . , n - l

equivalence

the

sarjection

.

to

η

ζ

when

i.e.

the n a t u r a l After of u n i f o r m

R

x^ = η

and t r a n s i t i v e and

R[ζ J

π : X + X/R,

»

.

is

bracketing

as

related

to

η

by

of

x^Rx^^^

points

for

if

R = R c R.

then

R

is called classes

is

if

is

s a i d t o be an

the equivalence

i s d e n o t e d by

g i v e n by

R

π(ξ)

= R[5J ,

X/R is

class and

called

projection. these preliminaries structure

The members o f the n o t a t i o n s

on a g i v e n

Ω on

for

all

This

such

the

definition

structure certain

relations

t h e o r y are s t i l l

A uniform X χ X

X.

are not c a l l e d

relation

(I . 1 ) .

set

square X χ χ s a t i s f y i n g

the f i l t e r of

we a r e r e a d y f o r

but

s t r u c t u r e on a g i v e n

set

that

ΔΧ c D

D t Ω ,

(ii)

D e Ω

implies

D ^ e Ω ,

D e n

implies

Ε « Ε c D

for

some

Ε e Ω.

is

a

conditions. (although

used)

(i)

(iiil

and s o

such

is

and

as t r a n s i t i v e

on t h e c a r t e s i a n

Definition

relations

a sequence

The s e t o f e q u i v a l e n c e

a filter

=

.

relation,

ς.

factors)

n,

symmetric

of

(n

x^ = ζ ,

We d e s c r i b e reflexive,

of

R"-related

an R - c h a i n o f

filter

Note t h a t

X.

i s unnecessary

X

ς.

= R[S[H]]

for e a c h s u b s e t

of

seme

that

(RoS)[H]

Thus

for

entourages.

X

is

uniform Λΐ! uLiuf c^ Note t n a t that

for

any

E" c D. is

to

(iii) η > 1

exists

this

to the

an e n t o u r a g e

condition Ε

such

that

e x t e n s i o n w i t h o u t comment i n what

follow.

uniform

structure

notation. uniform

on

structure

refine s

fi,

Ω = Ω'

is

Definition X an

s p a c e we mean a s e t X;

usually

Λ r e f i n e m e n t of

an e n t o u r a g e o f

is

there

We s h a l l u s e

By a u n i f o r m

set

e x t e n d s by i t e r a t i o n

Ώ'

X

alone

a uniform

In t h i s

is

sufficient

structure

Ω

coarsens

The d i s c r e t e

i s the structure

Ω of

is

a

Ω

is

s i t u a t i o n we s a y t h a t β'.

t o be e x c l u d e d we d e s c r i b e

(1.2).

together with a

such t h a t each entourage

Ω'.

or t h a t

X

If

Ω'

the p o s s i b i l i t y

the refinement

uniform

also

structure

as

that

strict.

on a g i v e n

i n which e v e r y s u p e r s e t of

the

diagonal

entourage. In t h i s

space.

s i t u a t i o n we d e s c r i b e

Clearly

o t h e r uniform Definition

the d i s c r e t e

structure.

(1.3).

set

X

sole

entourage. In t h i s

space.

uniform

filter

uniform

points

structure

uniform

s u p e r s e t s of

that the base g e n e r a t e s the

X

every

X

X χ χ

is

the

uniform

i s refined

has a t

by

least

and t h e

three

well.

so t h a t t h e members of the base,

In t h e c a s e

of

two

trivial

least

structures as

t h e members o f filter.

on a g i v e n

has at

structure

When

by a b a s e ,

set

structure

Provided

are d i f f e r e n t .

is given

uniform

refines

as a t r i v i a l

t h e r e are other uniform

a filter are the

X

uniform

structure.

the d i s c r e t e

structure

distinct If

points

structure

i n which the f u l l

the t r i v i a l

e v e r y o t h e r uniform

as a d i s c r e t e

At t h e o t h e r e x t r e m e we h a v e

s i t u a t i o n we d e s c r i b e

Clearly

distinct

uniform

The t r i v i a l

i s the structure

X

the

we s a y a uniform

I.M. James s t r u c t u r e we s a y t h a t t h e b a s e g e n e r a t e s and d e s c r i b e t h e members o f

t h e uniform

the base as b a s i c

structure

entourages.

For e x a m p l e t h e s y m m e t r i c e n t o u r a g e s a l w a y s form a b a s e , we s h a l l be u s i n g

frequently.

i s g e n e r a t e d by t h e Reversing for

a filter

our v i e w p o i n t , Ω

(1.1), (i)

as

on

X χ X.

s u p p o s e t h a t we have a b a s e For

ii

for

all

implies

Ε c

(iii)

D e 8

implies

Ε » Ε c D

Apart from t h e m o d i f i c a t i o n

for

to

For e x a m p l e ,

consider

d i t i o n s are s a t i s f i e d = { (ζ ,η) ,

sane

for

(ii),

a r e t h e same a s t h o s e on

Η χ Ε

Ω

some

therefore,

the

conditions

itself.

the r e a l

line

by t h e f a m i l y

of

JR.

The t h r e e

|ζ - η|
• D ,

since

for

a n o t h e r way of

uniform

preserving

« D",

and s o

There i s

n,

is

a symmetric entourage

such t h a t

some

λ

suppose that

Then t h e r e e x i s t s e X

all

= λ(η).

Conversely

ς,η

i s uniformly

is uniformly

space

: X -»· Ε

D = (λχλ)

for

X

uniform

λ

continuous function

consider

ε X

X

space

image s e n s e ,

The f u n c t i o n

as

i s uniformly

there e x i s t s

an e n t o u r a g e

Y,

open i f Ε

objects

structure-

follows.

φ : X

spaces,

the

of

where

X

for each Y

such

and

entourage that

Ε[φ (χ) ] c φ ( D C x ] ) for

all

X £ X.

Of c o u r s e i t for

sufficent

if

the condition i s

b a s i c e n t o u r a g e s when t h e u n i f o r m

by a b a s e . when *

is

Y

For example

is discrete.

φ

is

structure

of

always uniformly

In the s i t u a t i o n

of

(1.8)

T h e r e is an error in t h e corresDOndino result

satisfied X

is

given

open the (9.34)

condition of

[101.

for

φ

exists

t o be u n i f o r m l y an

η € Y,

e > 0

implies

open i s

such t h a t that

that

for

σ(φ{χ),η)

ρ(χ,ζ)

< δ for

each

< t,

some

δ > 0

there

where ξ 6 X

χ e X such

and

that

Φ (ξ.) = η . Note that its inverse continuous uniform

i s uniformly

(1.13).

continuous.

Let

and l e t

are uniform

and o n l y

Consequently

open i f

if

a uniformly

and o n l y i f

it

is

a

φ : X

ψ ; Y

spaces.

Ζ

If

Y

be a u n i f o r m l y

be a f u n c t i o n ,

ψφ : X

Ζ

continuous

where

X,

i s uniformly

Y

and

open

then

ψ . For l e t

is

open i f

equivalence.

surjection

so i s

i s uniformly

b i j e c t i o n i s uniformly

Proposition

Ζ

a bijection

Ε

be any e n t o u r a g e o f

an e n t o u r a g e o f

exists

X.

an e n t o u r a g e

If

F

of

ψφ

Y.

Then

i s uniformly

Ζ

such

is

uniformly

D = (φχφ) open t h e n

^E

there

that

F[ψφ (x) J c ψφΟ[χ] for

all

X £ X.

Then

ρ[ψ (y) ] c ψΕΓγ:; for

all

y £ Y,

Proposition ψ : Y ->· Ζ and

Ζ

so i s

and s o

(1.14).

ψ

Let

φ : X

be a u n i f o r m l y

are uniform

continuous

spaces.

If

be a f u n c t i o n injection,

ψφ

and

where

i s uniformly

let X, Y

open

then

φ . For l e t

D

be a n y e n t o u r a g e o f

open t h e r e e x i s t s an e n t o u r a g e Ε[ψφ(χ)] is

Y

open.

c ψφθ[χ]

an e n t o u r a g e o f

Suppose t h a t

for

all

Y ,

η £ Ε[φ(χ)]

F

points

since .

ψ Then

of

X . Ζ

If such

χ £ X .

is

uniformli

that

Now

i s uniformly ψ(η)

ψφ

Ε =

(ψχψ)~^Ε

continuous.

ε Γ[ψφ(χ)]

c ψφΟ[χ]

I.M. James and s o

ψ(η)

=

ψφ(ξ)

i n j e c t i v e we h a v e Ε[φ(x)]

c φΟ[χ]

Proposition where

so i s

some

η = φ(ζ) and s o

(1.15).

X, Y

uniformly

for

and

Ζ

φ

: X

open s u r j e c t i o n .

Since

η e φΟ[χ]

Y

ψφ

is

Thus asserted.

ψ : Y

spaces.

If

.

open, as

and

ψ

Ζ

be

functions,

Suppose t h a t

i s uniformly

φ

is a

continuous

then

ψ .

an e n t o u r a g e of Since

ψ

φ (ζ)

Ζ

Ε[φ(χ)]

e Ε[φ(χ)]

η £ D[x] this

and t h e n

Proposition function,

then

X Y

For s i n c e

for ζ

Let and

X e X

χ £ X .

then

So

Since

continuous,

φ : X

Y

for

an e n t o u r a g e

ψφ(ζ)

as

Ε

X .

of

Y

some = ψφ(η)

open

If φ

X

is

and

of

of

asserted.

spaces.

X

is

Y

is

surjective.

Φ such

is

uni-

that

= ΦΧ , Hence

η=1,2,... (φ{χ),η)

for

be a u n i f o r m l y

a r e uniform

Ε

F

if

φ (ξ) = φ ( η )

i s an e n t o u r a g e of

χ £ Χ .

we h a v e

£ X

Y

open t h e r e e x i s t s

Ε"[φΧ] c φχ

i s an e n t o u r a g e

i s uniformly connected then

X x X

points

all

i s uniformly

Είφ (χ) ] = φ( ( Χ χ Χ ) [ χ ] ) all

If

o p e n t h e r e e x i s t s an e n t o u r a g e

seme

(1.16).

n o n - e m p t y and

D = (ψφχψφ)"^Ρ

(ψφ (x) ,ψφ (η) ) £ F .

ψ

where

i s uniformly continuous.

c φο[χ]

for

shows t h a t

formly

ψφ

i s uniformly

such that

as

.

i s uniformly

are uniform

For s u p p o s e t h a t

for

and s o

φ

Let

ζ e D[x]

Ε [ φ χ ] c φχ

.

e e"

So i f for

seme

and

η £ Υ η

so then for

and s o

any

η £ φΧ

required. To c o n c l u d e

this

section

p l a y s an i m p o r t a n t r o l e mentioned pair

of

in the

transverse

X χ X ,

to each other i f

the diaqonal

ΔΧ .

concept

which

in the theory although I cannot find

literature.

s u b s e t s of

I introduce a further

F i r s t of a l l , for their

l e t us say that a

a given set intersection

More s p e c i f i c a l l v ,

it

X ,

are

i s contained

suppose t h a t

X

is

in a

Uniform structures uniform

s p a c e and t h a t

Let us

R

i s an e q u i v a l e n c e r e l a t i o n

s a y t h a t an e n t o u r a g e

D η R = ΔΧ . determined that φ

l:

R = (φχφ)~^ΔΥ than

entourage

is

,

φ : X

Y ,

R .

Note t h a t

of

s i n c e any s u b s e t

also

we s i m p l y

transverse.

X

to the

and o n l y

(1.17) .

where

Y

X ,

if

is

Let

and

Ζ

i s a uniformly

verse

to

ψ

then

(ii)

if

φ

i s a uniformly

then if

and

Y

φ Y

is

is

Φ(ζ)

e D

and

= φ{η)

,

one

to

transverse of

trans-

entourage

structure of

X

is

is

is

to the projection

continuous

of

to

to

ψ

where

Ε

hence X

X

(iii),

transverse

to

φ

and l e t

transverse

to

ψ ;

then

transverse

D Ε

Y

is

X

is

is

trans

transverse

to

which i s

transverse

transverse

to

to

ψφ .

i n t h e c a s e of to

(φ ( ξ ) , Φ ( η ) )

ψ;

ψφ ;

e Ε

and

let

D

then

E,

transverse

be a n e n t o u r a g e o f

(i),

if

(ii) ,

be an e n t o u r a g e of

D η (φχφ)~^Ε

t o ψφ .

is

Thus,

transverse

of

let

X

In t h e c a s e of

which i s

Y

ψφ ;

if

then

then

ζ = η.

i s an e n t o u r a g e

i n j e c t i o n and

is transverse

ψφ (ξ ) = ψ φ ( η )

be f u n c t i o n

ψ ;

continuous, to

Ζ

Then:

o p e n s u r j e c t i o n and

transverse

,

ψ : Y

spaces.

transverse

I n t h e c a s e of

which i s

so

transverse

consisting

and

are s t r a i g h t f o r w a r d .

a n e n t o u r a g e of (1.12),

is

transverse

D = (φχφ)~^Ε

(ξ,η)

in

X

i s uniformly

The p r o o f s let

of

structure

is

is a set, is

the uniform

Y

a r e uniform

φ

φ

Y

a transverse

transverse

φ : X

If

(iii)

of

say t h a t

it

(i)

ΨΦ

R

X .

point-space.

Proposition

to

t e r m when

D

a base

For e x a m p l e t h e u n i f o r m

if

if

where

to

the e x i s t e n c e

verse entourages, transverse;

R

and t h e n we s a y t h a t

implies the e x i s t e n c e

discrete

transverse

We s h a l l g e n e r a l l y u s e t h i s

by a f u n c t i o n

rather

D

on

X Y

to

so be as ψ .

which which

i s an e n t o u r a g e of

is is X

It

i.ivi.jomes By a l o c a l u n i f o r m

and u n i f o r m l y uniform verse

open f u n c t i o n

spaces,

to

.

The r e a s o n f o r

equivalences

Let

continuous functions,

Y is

are trans-

emerge

in

local uniform

By c o m b i n i n g

we o b t a i n

φ : X -»• Y

where

X

local

i s a l o c a l uniform e q u i v a l e n c e .

continuous

and

is also a

A l s o t h e c o m p o s i t i o n of

(1.18).

X

s t r u c t u r e of

equivalence

w i t h our p r e v i o u s r e s u l t s

Proposition

where

the terminology w i l l

Clearly a uniform

uniform e q u i v a l e n c e .

(1.17)

φ : X + Y ,

such t h a t t h e uniform

φ .

Section 3

e q u i v a l e n c e we mean a u n i f o r m l y

X, Y

and

and

Ζ

ψ : Y ->• Ζ

be u n i f o r m l y

are uniform

spaces.

Then: (i)

if

φ

i s an i n j e c t i o n ,

a l o c a l uniform

equivalence,

ψφ

then

i s uniformly ψφ

is

o p e n and

ψ

a l o c a l uniform

is

equiva-

lence; (ii)

if

uniform

i s a uniformly

φ

equivalence then

One f u r t h e r Proposition function, φ

(1.19).

where

φ

φ : X ->• Y Y

inverse

ψ

e n t o u r a g e of

Y

such t h a t

continuous. Υ . since

If Ε'

(Γΐ,φψ(η)) Ji(n)

is

η c Ε'[φ(χ)]

ε Ε ο Ε ,

= ΨΦΨ(η)

Ε ο Ε

Ε

is t r a n s v e r s e

to

an e n t o u r a g e of

and

Suppose

and l e t

Consider the entourage

f- F ,

continuous that

equivalence.

open.

b e a n y e n t o u r a g e of

^E

spaces.

is

which i s a l o c a l uniform

D

(φψχφψ)

local

equivalence .

be a u n i f o r m l y

a r e uniform

For l e t

F =

is a

w h i c h may be w o r t h m e n t i o n i n g

and

i s uniformly

and ψφ

i s a l o c a l uniform

ψ

Let

X

admits a l e f t

Then

result

open s u r j e c t i o n

,

where

X

Y ,

since

symmetric

ψ . is

Then

uniformly

Ε' = Ε η F η (ψχψ)~^0

χ e Χ ,

η « Ε[φ (χ) ] ,

φψ

be a

then

since

which i s

transverse

we o b t a i n t h a t

η = ΦΦ (η)

φψ(η) £ Ε [ φ ( χ ) ] ,

Ε ' c: Ε .

to .

ψ.

of

Hence

Since

But we a l s o

have

Uniform structures ψ (n)

e D[x]

,

since

η = ΦΨ (η)

£ ψΟ[χ].

uniformly

open,

uniformly

continuous

of

Y

for

this

i

(ψχφ)Ε'

c D ,

Therefore

as asserted.

Ε'[φ ( χ ) ]

so c φΟ[χ]

I n f a c t we o n l y

and t r a n s v e r s e

conclusion

and

t o be

need

t o the uniform

obtained.

and s o ψ

φ t o be

structure

is

2.

Induced and coinduced uniform

S u p p o s e t h a t we h a v e a f u n c t i o n a set

and

Y

(φχφ)

^E,

where

uniform

is

a uniform Ε

structure

φ

trivial

on

X,

if

Y

called

continuous.

structure

X.

structure

then

relation

determined

More g e n e r a l l y

Y

d e t e r m i n e d by t h e p s e u d c m e t r i c structure

σ

then

φ : X •>· Y

and

uniform

space

and

uniform

structure

is

transitive

ψ : Y X

and

Ζ Y

induced

by

X

are s e t s .

φ.

Or we c a n g i v e

structure

induced

by

ψφ;

the r e s u l t

X

uniform

is clearly

structure

. sense.

Ζ

X

the

^ΔΥ

uniform

We c a n g i v e

by

is Y

a the

the uniform uniform

the

same.

of

the

induced

ψ : Y

Ζ

property i s characteristic

structure.

Proposition where

has the

where

and t h e n g i v e

the

example,

in the following

induced

uniform

has

R = (φχφ)

σ(φχφ)

structure

The f o l l o w i n g

Y

has the

be f u n c t i o n s ,

ψ

for

has t h e uniform

d e t e r m i n e d by t h e p s e u d c m e t r i c

The a b o v e p r o c e d u r e Let

if

if

X

by t h e e q u i v a l e n c e

φ.

structure.

For another

structure generated

is

Y, g e n e r a t e a

structure

For e x a m p l e ,

then so does

X

images

the induced uniform

has the d i s c r e t e uniform

by

where

The i n v e r s e

as the c o a r s e s t uniform

i s uniformly

uniform

φ : X -»• γ ,

runs through the e n t o u r a g e s of

T h i s may b e d e s c r i b e d which

space.

structures

X,

(2.1). Y

the uniform i s uniformly

and

Let Ζ

structure

φ : X ->• Y

and

are uniform

spaces.

i n d u c e d by

ψ

continuous

if

Suppose t h a t

from t h a t o f

and o n l y i f

ψφ

be

is

Z.

uniformly

functions, Y Then

has φ

Induced and coinduced uniform structures continuous. For l e t for

Ε

be a b a s i c

some e n t o u r a g e

structure. is

If

ψφ

an e n t o u r a g e

of

(φχφ)~^Ε =

iently

uniform

i s uniformly

for

φ : X

Λ useful

special

φ

continuous

is

X

and

then

(ψφχψφ)

continuous,

is

^F

injective

as is

t o be u s e d . Y

injective

continuous

sufficient Let

X

and

(uniformly

X

suffic-

We d e s c r i b e

are uniform

and

asserted.

spaces,

has t h e

as

induced

injections

condition

φ : X ->• Y Y

i s given

left

embedd-

in

be a u n i f o r m l y

are uniform

continuous)

are unifonn

spaces.

continuous

Suppose t h a t

inverse.

Then

(>

is

is

an

embedding.

For i f

ψ : Y -»• X

e n t o u r a g e of

X

e n t o u r a g e of

Y .

then

is a left (φχφ)

The i n d u c e d u n i f o r m

structure

structure

is called

the r e l a t i v e uniform

is called t r a c e s on

to a subset

a s u b s p a c e of A χ A

φ : X ·• Y be t r a n s v e r s e

Let

be u n i f o r m l y to

Y .

t h e empty s e t and t h e

full

We p r o v e

X, Y

and

Ζ

be u n i f o r m

continuous functions If

φα = φ3

spaces.

Let

and l e t

φ : Y -»• Ζ

tlien t h e

coincidenci

I.M. James set

Μ = Μ(α,Β)

of

For

be

let

transverse a,Β

Ε

to

α

β

i s uniformly

open i n

a symmetric e n t o u r a g e of

φ .

Let

are uniformly

are entourages

and

D

be any e n t o u r a g e

continuous

of

X .

Y

the preimages

X .

such t h a t

of

X .

Ε Ε

Since

( α χ α ) ( β χ β )

Write

D' = D η (αχα)~^Ε η (βχβ)~^Ε If

ζ

α(χ)

c D'[u(x)] = Β(χ)

and

( α ( ζ ) , β (ξ))

where

to

φ ,

for

= φβ(ξ)

= β(ζ),

χ ε Μ ,

(2.6)

with

(1.16)

Let

X, Y

(2.7).

uniformly

connected.

functions

and l e t

φα = φ β

Let

and

,

then Therefore

and

i.e.

ξ

and s o

Μ

we Ζ

α, β : X

φ : Y

Ζ

Ε · Ε

ε Μ . is

is

Thus

uniformly

obtain be uniform Y

set

spaces,

be u n i f o r m l y

be t r a n s v e r s e

then the coincidence

to

with

X

continuous

Y .

Μ = Μ(α,β)

is

either

full.

The n o t i o n extended

α(ς)

φα(ς)

all

Corollary

empty or

u : Μ c χ

X .

By c o m b i n i n g

If

and

£ Ε, (β ( ζ ) , β (χ) ) e Ε .

However

hence

C Μ η D[x]

open i n

χ £ Μ

(α(ζ),α(χ))

£ Ε · Ε .

transverse C/tuix)]

,

of

induced uniform

to multiple

example of

this

of u n i f o t m

spaces.

with a family

procedure.

{iTj}

π . : nx , . The u n i f o r m

Specifically

of

projections,

course,

product

let

The c a r t e s i a n p r o d u c t

s t r u c t u r e on

which each of

uous.

We r e f e r

to

uniform

product,

and w r i t e

We h a v e t o

The u n i f o r m

of

{x^}

ΠΧ^

is

be

an

be a f a m i l y

cones

equipped

where

X. .

product

structure for

situations.

structure can,

ΠΧj,

show t h a t

Πχ^

is

the coarsest

these projections with t h i s

uniform

uniform

i s uniformly structure,

contin-

as

the

IIX^ = X . such a s t r u c t u r e e x i s t s .

For

this

Induced and coinduced uniform structures p u r p o s e we i d e n t i f y lI(Xj X X j ) cartesian

X x X

with the cartesian

by t h e o b v i o u s r e a r r a n g e m e n t of product

llDj

,

where

where

is

Dj

and w h e r e j. is

all

the image of

Dj .

Thus

also uniformly

TTj

i s not

index

of

Xj

for

under

each

HD^ ,

index

number o f

j

indices

the p r o j e c t i o n

only uniformly

j

product

cartesian products

but a f i n i t e

ΠΟ^

each

the

π^ χ

continuous

but

open.

In p r a c t i c e explicit

for

for

Then

The u n i f o r m

of

through the entourages

Dj = X^ χ X^

Note t h a t just

X χ Χ .

g e n e r a t e d by t h e f a m i l y

runs

factors.

D^ c x^ χ χ^

can be r e g a r d e d as a s u b s e t of structure

product

it

is

description

seldcm n e c e s s a r y

of

one p r o c e e d s by t a k i n g

the uniform

to refer

product

a d v a n t a g e of

back t o

the

structure.

the following

Instead

characteristic

property. Proposition Let

φ : A

and

nx^

(2.8). ΓΚ^

is

tinuous if

and o n l y i f

indexing

Δ

set,

Δ

and i t

left

is

of

Δ : X -I- X X X

let

ρ

of uniform is

φ

spaces.

a uniform

space

i s uniformly

the functions

For e x a m p l e ,

follows

φ^ = π^φ

con-

: A ->- Χ^

π^

for

X^ = X

power,

a t o n c e frctn

,

take

where

(2.8)

continuous. any

j,

that

J the

for

all

is

the

diagonal

In f a c t

since

we o b t a i n frcra

(2.2)

embedding.

product i s d i s c r e t e

family. is

J*^^

i s uniformly

of d i s c r e t e uniform

X

A

Then

the uniform

a uniform

an i n f i n i t e

Let

each of

inverse

The u n i f o r m family

where

product.

obvious.

Δ : X ->- X"^

has the

that

is

nXj = x"^,

function

be a f a m i l y

continuous.

The p r o o f Then

{Xj}

be a f u n c t i o n ,

t h e uniform

i s uniformly

j.

Let

spaces but generally

Note t h a t

uniformly

be a uniform

be a p s e u d o m e t r i c

in the case

open i f

space, on

the diagonal

X.

and o n l y i f

w i t h uniform It

of

a

finite

not in the

case

function X

is

structure

discrete. Ω,

i s a simple e x e r c i s e to

and show

I.M. James that the distance if

and o n l y i f

defined that

by

ρ

Ω

p.

is

Thus

Ωρ

Ω

where

Ω^

ρ

e n t o u r a g e of

section is

with r e s p e c t

for

Ω^.

Ω.

structure

Ωρ

structure

.

of u n i f o r m

For l e t

Then t h e r e e x i s t s such t h a t

a

D

such

Clearly

Ω

is

continuous.

by t h e c o l l e c t i o n

ρ

on

and

of

previous

such t h a t

a refinement Moreover

are

sequence

D^^ = D

X

of

{D } η

Ω^,

D e Ω Ρ

such uniform

struct-

be a

As we h a v e s e e n i n t h e

a pseudcmetric

^ i s uniformly

is generated

Ω.

η = 1,2,...

there exists

continuous

pseudometrics which

to

symmetric e n t o u r a g e s

a base for

vvtence

by t h e c o l l e c t i o n

runs through a l l

symmetric

c D^

t h e uniform

t h e c o a r s e s t uniform

i s generated

continuous,

of

is

of

i s uniformly

continuous.

uniformly

0^^,02» · · ·

ρ : X χ X -v ]R

a refinement

i s uniformly

In f a c t ures

function

and s o

structures,

Ω

as

asserted. The u n i f o r m : Xj ->• Yj Yj

product i s

functorial

be a J - i n d e x e d

are uniform

spaces.

family

of

in character.

Thus

let

X^

and

f u n c t i o n s , where

Then t h e p r o d u c t

function

Πφ^ : n X j ^ IlYj i s uniformly φ^

continuous

i s uniformly

if

e a c h of

the

functions

Thus t h e p r o d u c t f u n c t i o n

and o n l y i f

each of

the functions

is

a

φ^

is

equivalence.

The p r o o f s e x c e p t for

of

the following

the l a s t w i l l be l e f t

Proposition spaces.

and o n l y i f

continuous.

uniform e q u i v a l e n c e a uniform

if

(2.9).

Let

{Xj}

results to the

(2.10).

bounded uniform

Let

spaces.

{Xj}

ΠΧ^

and

reader.

be a f a m i l y

Then t h e u n i f o r m p r o d u c t

Proposition

are straightforward

is

be a f i n i t e

of s e p a r a t e d

uniform

separated. family

Then t h e u n i f o r m p r o d u c t

of ΠΧ.

totally is

imucea ana comaucea umjorm structures totally

bounded.

Proposition uniformly nXj

is

(2.11).

uniformly

entourage

.

(Cj,rij)

If n.

uniformly

family

for

some £ d"

as

let

and n^ ,

D = JlD^

is

where

the uniform

coinduced

we f i n d

Let

and

Ζ

the

situation

φ : X ·> Y

are uniform

structure coinduced

i s uniformly

continuous

X^

.

for

Thus

X

if

is

rather

less

ψ : Y

spaces. φ

than

Ζ

is

be

functions,

that

from t h a t o f ψφ

to

straightforward.

Suppose

and o n l y i f

without

When we t u r n

the dual c a s e

and

by

structures,

c a n be d e f i n e d

structures

property required in

(2.12) .

of

η = max(nj)

structures,

and s e e k

The c h a r a c t e r i s t i c

basic

η = (η.) a r e p o i n t s of X 3 , since X^ i s u n i f o r m l y

t h a t induced uniform

induced uniform

Y

product

asserted.

therefore,

structures,

X,

non-empty

be a

an e n t o u r a g e

and h a v e t h e e x p e c t e d p r o p e r t i e s .

Condition

of

Then t h e u n i f o r m

D^

)

(ζ,η)

the dual q u e s t i o n ,

ψ

ζ = (ξ

so

We c o n c l u d e ,

where

example, where

connected,

and m u l t i p l e

be a f i n i t e

spaces.

e D^

and

difficulty

for

X = ΠΧ^ , j

connected,

induced

{X^}

connected.

(2.11),

of

each index

is

Let

connected uniform

To p r o v e

then

2i

Y

X.

has

Then

i s uniformly

con-

tinuous . To f u l f i l l "finest"

uniform

Consider, entourages

i n any o f

uniformly

of

this

for

t o g e t any i n s i g h t

of

the uniform

φ

structures

a uniform

At t h i s

level

i n t o the nature

of

Y

i s uniformly

s u b s e t s of

Take f i n i t e

as a b a s e for

satisfied.

which

the family

continuous.

fam i l y is

structure

therefore,

is

(2.12)

t h e c o n d i t i o n one must t r y t o g i v e

Y χ Y

on

Y

the continuous. which are

for which

Φ

intersections

of

members

structure

Y;

then

on

of g e n e r a l i t y the

it

structure.

is

hard

I.M. James For our p u r p o s e s , We s h a l l φ χ φ

however,

the general

of

the entourages

obviously

of

necessary that

Let

R

and l e t

X

D

of

X/R

X

D'

be t h e

there e x i s t s

the images,

form a b a s e

on t h e u n i f o r m

an e n t o u r a g e

for

under

a uniform

π χ IT ,

structure

with this

space

X ,

D'

such

X

if

that for

that

of

on

t h e e n t o u r a g e s of

X/R,

such t h a t

as the q u o t i e n t structure,

X

(2.12)

uniform

as the

quotient,

space.

Examples where a subspace

A

of

(2.13) X

b r e a k s down can be g i v e n by

and s e t t i n g

o b t a i n e d by c o l l a p s i n g = (1,1)

(0,e/2)

,

and

equivalent

,

e

and s o n o t t o

(l,e/2)

Definition X

entourage

is

D

to a point.

D = U,^ .

so that (e/2,1

-

(e/2, e/2)

, so that

Specifically

Then i f U^ ,

choosing

1 -

take

0 < e < i

and

(0,e/2)

s/2)

X/R

both is

belongs

does n o t be l o n g t o

to U, "2

R » U;^ R . however,

t o o weak,

(2.14). is

R = Δ υ (ΑχΑ)

belong to ,

whereas

It turns out, (2.13)

A

and t a k e

(l-e/2,1)

to

•> R ο U

space

is

surjective.

of

X/R,

in

exoimple i t

the uniform s t r u c t u r e

s t r u c t u r e and t o

e

for

Y .

L e t us s a y

to this

U

under

natural projection.

We r e f e r

(X,A)

of course;

relation

satisfied.

is

essential.

0 R 0 D' c R ο D ο R .

In t h i s c a s e

uniform

not

form a u n i f o r m s t r u c t u r e on

s h o u l d be

b e an e q u i v a l e n c e

π : X

(2.13)

is

φ

i s weakly compatible w i t h

each

is

o n l y be c o n c e r n e d w i t h s i t u a t i o n s where t h e images

T h i s d o e s n o t happen a u t o m a t i c a l l y ,

R

case

t h a t for most purposes the

and i t

i s usually replaced

The e q u i v a l e n c e r e l a t i o n

c o n p a t i b l e w i t h the uniform of

R ο D' c D » R .

X

there e x i s t s

R

structure

an e n t o u r a g e

D'

condition

by t h e

following.

on t h e

uniform

if

for

each

such

that

nduced and coinduced uniform structures This implies

(2.13),

inverses

the

with

•> R C R = D

is

D'

sufficient

basic a

stated

for

Note t h a t

is equivalent

instead

of

condition

R » D'

by

taking

to the

condition

c D " R.

t o be s a t i s f i e d

when t h e u n i f o r m

structure

Clearly

it

i n the case

of

of

X

i s g i v e n by

base. For e x a m p l e ,

where

Y

and

relation

(φχφ)

uniformly

^ΔΥ

then

π

on is

is

X

If

and X/R

open,

on

X/R

Suppose t h a t entourage entourage

of

X.

D'

R

is

has

of

uniform

with

the

space

is

r e l a t i o n on t h e

the q u o t i e n t

uniform

and u n i f o r m l y

π : X ->· X/R

is

compatible

with the

is

to a given

compatible.

Then

D'

X .

We h a v e

Let

« R c R ο d

(ιιχπ)0'[ΤΓ ( x ) D = π ( ( 0 · " Κ ) [ χ ] )

uniform

uniform

uniform structure

open.

uniformly

structure

the quotient

R

equivalence

compatible

t h e uniform

is

is

b e an e q u i v a l e n c e

with respect

compatible with

structure

R

continuous

projection

φ

projection,

Y.

Let

X .

left

Then t h e

and t h e q u o t i e n t

to

uniformly

the natural uniformly

structure

be t h e

spaces.

d e t e r m i n e d by

(2.15).

space

structure

φ : Y χ Τ -»· Y

are uniform

equivalent

Proposition uniform

let

Τ

product uniform

R

course.

condition

the

entourages,

of

on

Conversely

continuous

and

structure,

then

X

uniform

and t h e

structure. D

for

be a

some

symmeiric

symmetric

= w((R»D)[x])

= w(D[x]) for

all

X c X.

the quotient

Since

uniform

(τΓχπ)Ο'

structure

i s an e n t o u r a g e o f

this

shows t h a t

ττ

is

X/R

in

unifonrily

open. Conversely uniformlv

open,

suppose with

that

respect

π

is

uniformly

t o some u n i f o r m

continuous structure

on

and X/R,

i)

26

I. Μ. James

If

D

is

a symmetric entourage

of

X

then s i n c e

π

is

uniformly

o p e n we h a v e E[m (x) ] c: π Ο [ χ ] for

some s y m m e t r i c

entourage is

of

entourage

X/R

since

an e n t o u r a g e o f

X ,

by u n i f o r m

the

lence that

in

continuity,

the

relations

R

and

S

X/R

and s o

.

D = (wxtr)

and t h e n

(irxiTjD

is

an

On t h e

o t h e r hand i f

^E

an e n t o u r a g e

Ε =

for

forming

is

(πχπ)Ο

following

q u o t i e n t uniform

sense.

on t h e u n i f o r m

are weakly

Let

space

is

compatible, to

relation

.

This

Ε of

completes

by

As we h a v e a l r e a d y section

R

X .

of

the

structure.

(1.13)

seen,

for

entourages

In f a c t

the

For i f

S/R

sane entourage

D'

is

is

X,

equiva-

R c s, is to

. is

so

defined. X,

If,

S/R

is

then

further,

compatible

(2.13).

any u n i f o r m

space

X

the

an e q u i v a l e n c e

compatible with

any e n t o u r a g e

of

be

X/R

X/R

then

and

Ξ

respect

to

X

and

on

constitutes

relation D

to

structures

such that

compatible with

with respect X/R ,

R

X

i s weakly compatible with respect

with respect

for

then

an i n d u c e d e q u i v a l e n c e

S/R

on

of

proof.

transitive

S

Ε

Ε • Ζ

spaces.

that

of

is

be

functions,

Suppose t h a t

Y.

Then

injective

is

terminology. if

ψφ

is

the con-*

sufficiently

We s a y φ

is

that

injective

and

topology. subsets

of

topology,

space

topological

Specifically, X

spaces are

usually called

the

if

A

then the c l o s u r e

the intersection with

A

of

always

relative is

in

a subset

A

of a

the closure

of

subset of

Η

X

is

X. L e t u s now r e t u r n t o t h e u n i f o r m

uniform

space

is defined

a t o p o l o g y on

by t a k i n g

each subset D

trivial

continuous.

φ

special

in t h i s case.

A

and

a t o p o l o g i c a l embedding

the t o p o l o g i c a l Η

ψ

c a s e when

In p a r t i c u l a r , given the

Y

i n d u c e d from

has the induced

topology

the

).

are t o p o l o g i c a l

to deserve

φ : X ->- Y

(2.1

φ : X

and o n l y i f

The s p e c i a l

X

topology is

The i n d u c e d t o p o l o g y h a s t h e f o l l o w i n g

property,

where

the induced

Η

of

X,

called

the c l o s u r e X,

i s the

where

R = nD

but g e n e r a l l y

uniform

t h e uniform structure If metric

structure

is

topology

the t r i v i a l

where

Η

contains

entourages,

with

topology,

with the t r i v i a l

the

while

uniform

topology.

s t r u c t u r e on

t h e n t h e t o p o l o g y on

w i t h t h e uniform

the

associated

the d i s c r e t e

topology associated

is

of

for

subset.

the uniform

the uniform ρ

the i n t e r s e c t i o n

as a proper

For e x a m p l e , discrete

is

H,

η DCH],

Of c o u r s e

a

topology,

i n which

intersection X.

If

t h e unifor m

operator

runs through t h e e n t o u r a g e s of

R[H],

category.

X X

i s defined

by a p s e u d o -

d e t e r m i n e d by

p

toDoloav a s s o c i a t e d w i t h the uniform

coincides structure

The uniform topology d e t e r m i n e d by

ρ.

Topologically

Different t o the line

discrete

uniform

but not uniformly

structures

same t o p o l o g y . E,

33

of

w h i c h i s g e n e r a t e d by t h e f a m i l y ΔΕ υ ((α, k

which the

continuous.

cannot contain

(k+2)/2"]

establishes Perhaps

S t

c a s e when

.

η

α(η)

,

is

α

uniformly

a(n)

S

and

for

and s o

that

I assert

ε Ε [ χ ] => η ε α(ζ)

t

Hence,

special

with

= 0 .

- α (η) I < that

,

: ζ ε E^Lx]} a(x)

is

in the

t o be t h e g r e a t e s t

and

E^^ = ΔΧ ,

implies

implies

η

k/2"

D^^ = E^ • Y

such

that

unique.

at a given point

The n e i g h b o u r h o o d f i l t e r

A,

Therefore

that

i s a ccmplete

a separated complete uniform

Cauchy c o n d i t i o n filter

the

ccmplete.

φ : A

To d e f i n e follows.

on

hypothesis.

proposition

is

of

an a d h e r e n c e p o i n t

The same a r g u m e n t s h o w s , more g e n e r a l l y , ccmplete

be a Cauchy

a Cauchy f i l t e r

hence

and h e n c e a l i m i t p o i n t o f X/R

F

be an u l t r a f i l t e r

and s o a d m i t s a l i m i t p o i n t

a limit

Therefore

Let

and s o t h e t r a c e (4.7).

Since

χ N^

of

X

of

we p r o c e e d

χ

satisfies

of

on

A

φ

i s uniformly

is

as the

a Cauchy

continuous.

Completeness and completion t h e image o f

the trace

Y

is canplete

y

i s unique

is

Y

is

N e x t we show t h a t on

A.

since

φ

is

neighbourhood V e G^, ψ (x)

of

U

since

of

of

Y

x'

D

η Ν'

is

N'

of

Φ V

is

a n e i g h b o u r h o o d of

χ

in

for

φ (UnA) c V

G^

converges

sane

and so to

Ν

contained

into

ψ

on

A

y

and

Ε .

X

F.

F,

be a n y that

X that

This w i l l

establish

to

(x ,x ')

x, x ' ,

Thus f o r

any

But s i n c e

M' £ F' Φ*F

φΝ ^ Γ ί ψ ( χ ) ]

φΝ' c Γ [ ψ ( χ ' ) ] points

£ D.

respect-

Μ T F,

we h a v e t h a t

Similarly

so

of

I assert

φΜ x φΜ' c f .

the

D

such that

converging

and s o

F'.

Ε

for

ξ £ Μ η Ν

pairs

(φ(ζ·),ψ(χ·))

that

£ F ο F " F c Ε ,

required. Finally,

so

X.

(φ(ζ),ψ(χ)), in

F.

open t h e r e e x i s t members

of

of

let

an o p e n e n t o u r a g e

F •> F ° F

on

we h a v e t h a t

( ψ ( χ ) , ψ ( χ ' )) as

= y .

be a s y m m e t r i c e n t o u r a g e s u c h

by d e f i n i t i o n

(φ(ς),Φ(ς')), are a l l

χ e Α.

continuous,

be p o i n t s o f

ψ (x)

s a n e member

£ Μ'

F

into

Μ X M' ^ D

some member ς'

D

be f i l t e r s

converges to

and

If

Then

D η (AxA)

F,

such that

for

maps

X,

Since

Ψίχ)

φ ^V = U η A

Thus

There e x i s t s

So l e t

ively.

X.

i s uniformly

continuity

F'

in

and l e t

maps

the uniform

Let

ψ

φ χ φ

ψ χ ψ

is

G^

.

F ο F ο F c E. such t h a t

χ

of

'Since

c o i n c i d e s with

where

and s o

φ (UnA) c G^.

To show t h a t entourage

φ '''V

Y.

We d e f i n e

y = φ(χ),

then

on

y e Y

thus d e f i n e d ,

continuous,

= y = φ (X)

then

ψ,

y

G^

a limit point

separated.

For s u p p o s e t h a t

a neighbourhood A,

a Cauchy f i l t e r

there e x i s t s

since

61

we h a v e t o e s t a b l i s h u n i q u e n e s s .

In f a c t

if

and

I.M. James ψ'

is

a continuous extension

continuous, since

Y

then

is

Corollary

( 4 . 2 2) .

ively.

If

uniformly

Aj^

is

extension

of

respect-

equivalent

and s o

c a n be c o n s t r u c t e d

P,.^

X^, then

X^

f

: X^^

X2,

extends

by

gf

gf

= id

is

,

g : X j "*"

: X^^ -»• Xj^ id^^

is

on

·

an

X^

is

also

Similarly

equivalence,

the c o l l e c t i o n

a

while

by u n i q u e n e s s .

a uniform

to

(4.21),

continuous function

The i d e n t i t y

f

02»

If

as

asserted.

of Cauchy f i l t e r s ,

are c a l l e d m i n i m a l Cauchy f i l t e r s . as

is

the inclusion.

then

continuous function

X,

to

denote

equivalence

t o a uniformly

space

X^ ,

•

Minimal e l e m e n t s of jniform

X.

s u b s e t s of

to

Therefore

on

A,

be d e n s e

σ^^α ^α = σ^ .

.

X2

on t h e c l o s u r e of

uniform

a uniform

s u c h an e x t e n s i o n

uniformly

be s e p a r a t e d c c m p l e t e

i s uniformly

Now t h e u n i f o r m l y

necessarily

X^ ,

continuous function

extends

not

coincide

σ^ : A^ - • X ^ ( i = l , 2 )

A2

uniformly

φ,

and s o c o i n c i d e

Λ2

equivalent

α : Aj^

ψ'

Let

A^^ ,

For l e t

Eg = i d

and

Hausdorff,

s p a c e s and l e t

Oj^a ^

ψ

of

on a

These

follows.

Proposition

(4.23).

space

Then t h e r e e x i s t s o n e and o n l y o n e minimal Cauchy f i l t e r

FQ Fq

X.

on

X

Let

such that

is given

F

f

i s a refinement

by t h e s u b s e t s

through t h e e n t o u r a g e s of of

be a C a u c h y f i l t e r

DCM]

of

and

Μ

X

of F,

on t h e u n i f o r m

FG. where

A base D

for

runs

r u n s t h r o u g h t h e members

F. As i n

FG

on

Fg

is

X.

(4.16), If

Μ

of

i s D-small

then

a Cauchy f i l t e r

the proof

suppose t h a t

G i v e n an e n t o u r a g e exists

the family

D

a D - s m a l l member

subsets

D[M] g e n e r a t e s

D(;M]

X,

G

is

a Cauchy f i l t e r

X

and a member

Ν

of

G

and

coarser

Ν

filter

( D » D » D ) - s m a l l and

on

of

clearly

is

a

Μ

than

F.

refined of

meets

by

so

To c o m p l e t e F.

F

there

M,

since

Completeness and completion Ν e F,

hence

a refinement

Ν c D[MJ of

Fg

members o f

F ,

For e x a m p l e

take

given point filter

FQ

F^

it

is

is

χ

Then

and s o we Let

N^

is

X

sufficient

to use only

i s generated

F^

'G

is

is

basic

by a g i v e n

filter

base.

generated

just the

by a

neighbourhood

obtain be a u n i f o r m

space.

a m i n i m a l Cauchy f i l t e r

for

The

neighbour-

all points

χ

X.

Corollary uniform

( 4 . 2 5) .

space

interior

of

Let

X.

If

is

Μ

be a m i n i m a l Cauchy f i l t e r is

an e n t o u r a g e

an o p e n e n t o u r a g e ELMj

F

a member o f

open s i n c e

illustrates

D

Ε c D. Ε

Our n e x t r e s u l t , one of

of

If

A;

of

any

if

F

all

i s open,

Proposition relation

X

the the

of

(3.17) ,

E[M] = DLM]

and

result. (4.2U)

above,

filters.

a uniform

space

t h e m i n i m a l Cauchy

t o obtcjin

X

and

of

G

on

information

A

not

filter

is

defined F

about

fran

tlie

trace. Let

on t h e u n i f o r m

is

the

by

w h i c h c o n t a i n s members w h i c h do

then the t r a c e

the

then

whence

A

t h e manbers of

(4.26).

lence c l a s s e s Then

X

i t may be p o s s i b l e of

there e x i s t s ,

w h i c h may b e c o m p a r e d w i t h

on

A

meet

properties

then so i s

t h e u s e s of m i n i m a l Cauchy

F

a Cauchy f i l t e r

X

Μ £ F

S u p p o s e t h a t we h a v e a s u b s p a c e

meet

F

on

M.

For g i v e n

G

Therefore

minimal.

t o be t h e p r i n c i p a l X.

(4.24) .

hood f i l t e r of

F of

of

Corollary

D[MJ e G.

when t h e f i l t e r

χ

Njj

and s o

and s o

In c o n s t r u c t i n g

63

R[x]

R

space

be a c c m p a t i b l e X,

i s ccmplete.

such t h a t

equivalence e a c h of

Suppose t h a t

the

X/R

equiva-

is

ccmplete.

ccmplete.

For l e t t o some p o i n t

F

be a Cauchy f i l t e r

τιίχΐ

4 X/R .

on

I assert

X.

Then

w^F

converges

t h a t e a c h member o f

t h e minimi

I.M.James Cauchy f i l t e r G

G

trace

of

on

argue

that the

R[x]

is

d e t e r m i n e d by R[xJ

trace

Then

a limit point

of

ς

point

is

a limit

meets

is defined. converges

complete.

F

G

ζ

of

so t h a t

Assuming t h i s

t o some p o i n t is

itself,

R[x],

an a d h e r e n c e

since

G

we g o on

F

of

to

ζ e R[x],

since

point,

hence

and

i s Cauchy.

the refinement

the

Therefore

G,

and s o

X

is

complete. To p r o v e R[x]

for

t h e a s s e r t i o n we h a v e t o

each entourage

D

By c o m p a t i b i l i t y

we h a v e

E.

converges

Since

Ti

a neighbourhood member

Ν c Μ

DCN] C D[M] completes

of

The m a i n u s e ccmpletion

the uniform If,

of

for

it

Cauch^'filters

X

Ν c

X/R

uniform

of

as f o l l o w s .

i s not

the r e s u l t

space on

pairs

such subsets on

0^(1=1,2)

entourage

for

c D~^[R[x]]

the a s s e r t i o n

is

in

is some

and

so

and

is

X X,

(F,G)

D*

of

instead, in of

but

general. the

t h e same a s i f

separated

minimal

place.

l e t us consider with

the

construction

separated,

by t h e c o n s t r u c t i o n

For e a c h symmetric

structure

F.

ττ(Ε[χ])

A similar

the s e t

the following

entourage

D

of

X

X χ X

X

I assert

forms

of

uniform let

D*

o f m i n i m a l Cauchy f i l t e r s that

a base for

on the

a

X.

T h a t t h e fam i l y f o r m s a b a s e f o r For l e t

some and

minimal Cauchy f i l t e r s

space

of

of

πΝ c TI(EL'XJ)

w h i c h h a v e D - s m a l l members i n corunon.

family

for

(RoE)[x]

were used i n the f i r s t

set

Μ

(4.26).

minimal Cauchy f i l t e r s

the

in

This proves

i s followed

Given a uniform

denote

«R

D[M] m e e t s

out u s i n g o r d i n a r y Cauchy f i l t e r s

uniform

structure.

π (x)

space which a r i s e s

however,

quotient

Then

construction,

can be c a r r i e d

to

RLxJ.

t h e proof

and member

we h a v e t h a t

F.

meets

X

R o E c D * ^

Π{χ)

of

of

show t h a t

be a s y m m e t r i c

a filter

entourage

of

i s obvious X.

The

enough. inter-

Completeness and completion section

D = D^^ η D^

D-small

s u b s e t of

D* c D* η

X

as

The f i r s t

is also is

two o f

a uniform

is

a Cauchy f i l t e r

the three

structure on

are X

then

D*

is

s y m m e t r i c by

that X

G,

X,

and l e t Let

such t h a t both

completes

Ε

F,

(F,G)

Now

G,

on

is

the proof

D

of

In f a c t

such t h a t X.

and

G.

Now

H

F

(F,G)

minimality,

X

is

function

G,

i

D*

e D*

Μ u N, Η

: X - 0 .. ζ = (£/4,0)

G

take

is

is

filter

ccnrnutati-ve.

G = JR, κ Η

w i t h t h e m u l t i p l i c a t i o n g i v e n by

( x , y ) · ( x ' , y ')

where

in

satisfying

g ^Vg c W f o r

For an e x a m p l e w h e r e t h e y a r e d i f f e r e n t

Η

G

G

W .

The u n i f o r m

Take

character-

V £ Ν .

two w a y s .

the r i g h t uniform let

on

some

t o take the right r e l a t i o n s

is called

a group

Ν

are s a t i s f i e d

in e i t h e r of

relations

The o t h e r i s

group i s

W £ W then

conditions

uniform s t r u c t u r e

the s e t

the l o c a l

introduce

conditions:

if

b a s e for

through

together with a f i l t e r

(1)

sufficient

H\G .

follows.

the f o l l o w i n g

this

and

G

t h e m o s t a p p r o p r i a t e way t o

topological

algebraic sense

the l e f t

i n the group

.

g e n e r a t e d by t h e

: ix-1I < £,

|y|

< £}

With ,

η =

(£/4,£/2)

subsets

t o be

I.. James we h a v e Π.ς"^ = ( 1 , £ / 2 )

Thus t h e

left

ε U^ ,

= (1,2)

r e l a t i o n d e t e r m i n e d by

the r i g h t r e l a t i o n determined So t h e left

r i g h t uniform

uniform

ment of

of a s u b s e t

subsets

H.W ,

where

a base for

t h e members o f

r i g h t uniform the closure

structure, Η

i n t h e c a s e of

the

the

case.

so t h a t

bourhood f i l t e r of

function are

of

for

ε > 0 .

left

any

of

the

i s not a r e f i n e -

observe G

t h a t the uniform

are generated the

Similarly

with

W.H

i n the uniform left

uniform

left

topology of

e,

topology

structure

the given while

uniform Η

or

of

H.W.

η(W.H) these

the

Thus ii(H.W) in

the

topologies

t h e same i n

filter

either

W becomes t h e

is

the

neigh-

neighbourhood

C l e a r l y both the

and t h e m u l t i p l i c a t i o n

the

i s g i v e n by and

is

gAJ = Wg

g ε G.

of

In f a c t

topology

by

i n t h e c a s e of

in p l a c e

structure.

the uniform

any e l e m e n t G •» G

H

inversion

function

G χ G -·· G

continuous. Of c o u r s e t h e p r o c e d u r e

a topological

group

group s t r u c t u r e functions e

the

case,

Ν .

r i g h t uniform

In t h i s

filter

contain

W r u n s t h r o u g h t h e members of

of

coincide,

does not

not a refinement

i n t h e c a s e of

structure,

c a s e of

is

similarly

to the general

neighbourhoods of

U^

U^ .

right.

Returning

family

structure

structure,

the

by

Uj^

/

(i)

with

in which the i n v e r s i o n

-

(iv)

topology.

We may d e f i n e

space carrying

and

a

multiplication

Then t h e n e i g h b o u r h o o d f i l t e r

so that the l e f t

can be d e f i n e d ,

w i t h the uniform

reversible.

as a t o p o l o g i c a l

are continuous.

satisfies

structure

G

is

and r i g h t

and t h e o r i g i n a l However,

it

topology

i s more

of

uniform coincides

consistent

our a e n e r a l approach t o r e g a r d t h e t o p o l o g i c a l

structure

Topological groups as d e r i v e d

from t h e u n i f o r m

structure,

r a t h e r than t h e

Other

way r o u n d . The l e f t uniform

and r i g h t u n i f o r m

structures

groups.

considered

Another s t r u c t u r e

s e c t i o n of

the l e f t

fact

in the

of

t h e o r y of

some i m p o r t a n c e

and r i g h t s t r u c t u r e s ,

two-sided structure. uniform

s t r u c t u r e s are not t h e

However,

only

topological i s the

inter-

sometimes c a l l e d

t h e p r o p e r t i e s of t h e s e

s t r u c t u r e s w i l l n o t be d i s c u s s e d

in t h i s

other

book.

In

we s h a l l g e n e r a l l y c o n c e n t r a t e on t h e r i g h t u n i f o r m B e f o r e g o i n g any f u r t h e r

few e x a m p l e s .

Of c o u r s e ,

filter

structures Again,

any g r o u p

taking

g e n e r a t e d by

G

g r o u p by t a k i n g

e;

both l e f t

consisting

and r i g h t u n i f o r m s t r u c t u r e s is

t o be

and r i g h t

sense, the

uniform

discrete.

of

G

itself;

both

left

and t h e u n i f o r m

trivial.

b" ( n = 0 , 1 , . . . )

c o n s i d e r the r e a l

with the a d d i t i v e group s t r u c t u r e ,

i s the n e u t r a l element. t h e open t - b a l l s positive reals.

The u n i f o r m

Ν

topology is

are t r i v i a l

For a c o m m u t a t i v e e x a m p l e ,

structures

in the a l g e b r a i c

We t a k e

centred at zero, In t h i s c a s e

a r e t h e same, s i n c e structure

sidered e a r l i e r ,

is

just

n-space in which

W t o be t h e f i l t e r where

the l e f t

«

the group i s

topology

the

uniform

commutative.

the Euclidean structure

and t h e u n i f o r m

g e n e r a t e d bi

runs through

and r i g h t

zero

is

j u s t the

conEuclidean

topology.

For a non-commutative example, take the general linear group

Ga(nJR)

base for arouD bv

a

can be r e g a r d e d a s a t o p o l o g i c a l g r o u p by

W t o be t h e f i l t e r

topology

G,

a r e d i s c r e t e and t h e u n i f o r m

any g r o u p

structun

w i t h t h e t h e o r y l e t us c o n s i d e r

can be r e g a r d e d a s a t o p o l o g i c a l principal

the

of automorphisms o f

By choosing a.

we c a n r e p r e s e n t t h e e l e m e n t s of (nxn)-matrices , and h e n c e t o p o l o g i z e

the it a s a s u b s e t

I.M. James of

Then

η = 1

GZ ( η , Β )

t h e g r o u p c a n be i d e n t i f i e d

group of

real

and t h e

numbers.

left

Thus c o n s i d e r where

T

t h e group

is

topology of of

a uniform

entourages Wj^ =

{0

of :

T,

(i)

consists

of

t W^,

i.e.

uniform

of

space,

with

consists

ε D

(iv)

pairs

is

again,

non-commutative

distinct. illustrations.

t h e uniform

of

multiplicative

equivalences

θ : τ

structure

topology, the

the subsets

i.e.

Τ, of

the

neighbourhood W^,

for

all

topological

of

for

group s t r u c t u r e s

R = R,,

e l e m e n t s of

(ψφ"^(t),t)

e D

just

e Τ} .

relation

(Φ,Ψ)

all

arises

self-equivalences

G for

such

that

all

t,

Thus t h e

left

rather

of

uniform

we form

are

determined

structure

the if

e D

s.

the uniform

but in g e n e r a l

The same s i t u a t i o n

topological

for

such t h a t

structure

are

a base for

Vt

The r i g h t

(ψ(s),φ(s))

convergence not.

structures

When

where

-

readily verified.

the

the group i s

G of u n i f o r m

(e(t) ,t)

The c o n d i t i o n s

,

In t h e a s s o c i a t e d

identity

D

such that

η a 2

uniform convergence,

the

with

p r o v i d e some i n t e r e s t i n g

uniform convergence.

filter

When

and r i g h t u n i f o r m

Function-spaces

Wj^

becomes a t o p o l o g i c a l group.

G

by

i.e.

right uniform structure

from t h e

than uniform

is

group

self-

equivalences . For a n o t h e r bijections Regarding uniform

Τ

: Τ

i.e.

T,

consider

where

Τ

is

as a d i s c r e t e uniform

structure

topology, for

0

illustration

of

an i n f i n i t e

t h e t o p o l o g y of p o i n t w i s e filter

of

the

subsets : θ (t)

= t V t ε S)

,

G

In t h e

convergence,

identity

of

set.

space l e t us g i v e

pointwise convergence.

the neighbourhood

Wg = {Θ

the group

G

the

associated a base

c o n s i s t s of

the

Topological groups where

S

runs through the f i n i t e

conditions

for

verified.

a topological

The l e f t

of

T.

Again

group structure

are

readily

relation

subsets

L = L

d e t e r m i n e d by

W_

WG

consists (t) all

of p a i r s = t

for

t e S.

uniform

(Φ ,ψ)

t € S,

i.e.

Thus t h e

left

uniform

The c l a s s

of

of

pointwise

invariant

neighbourhoods. group i f

neighbourhoods fact

it

is

V

since

V

if

Obviously a l l

the group

R* Κ R

groups s i n c e

for

the

left

and

interest.

they admit

a topological

small

group

G

admits a base c o n s i s t i n g

^ = V

for

all

U

of

gVg ^ c U

the conjugates

for

gVg ^

g t G . Ν

In

there

all

of

exists

g e G ,

for

all

g e G

condition.

abelian groups are

groups with d i s c r e t e

just

which the

e a c h member

such t h a t

the previous

is

= φ(t)

i s of c o n s i d e r a b l e

Ν

gVg

for

that

ψ(t)

structure

Specifically

such t h a t

t h e n t h e u n i o n of

fulfills

such t h a t

SIN

and o n l y i f

sufficient

a neighbourhood

these

such

convergence.

structures coincide

[ 2 1 ] we c a l l

SIN

G

t o p o l o g i c a l groups for

Following

i s an

S

e l e m e n t s of

all

structure

r i g h t uniform

of

the

or t r i v i a l

considered

SIN

groups,

topology.

earlier

as

are

On t h e o t h e r

i s n o t an SIN

hand

group.

We h a v e Proposition group,

(5.2).

Let

G

in the r i g h t uniform

be a t o t a l l y bounded structure.

Then

topological

G

is

an

SIN

group. For l e t V.V.V"^ c υ have

U ί .

V.S = G

By ( i v ) where

of

be g i v e n

Since for

(5.1)

g £ V.g^

is

and l e t

V c Ν

t o t a l l y bounded,

seme f i n i t e

we h a v e

j = 1 , . . . ,n

we h a v e

G

,

subset

gj.W^.g^"^

c ν

for

Then i f

W = W^^ η . . .

for

some

and

gW.g"^ c V . g j . W . g " ^ c

such

so

V.V.V"^ c U ,

that

on t h e r i g h t ,

S = {g^^ , . . .

.

j

he

seme η W^

of

we G .

W^ e W , and

g £ G

76

LM. James

as r e q u i r e d .

It follows,

bounded on t h e r i g h t and v i c e v e r s a ; Returning

it

then is

left

v i o u s l y a uniform

is

right p^

e

g ^Vg c W.

say that let

Thus

V

likewise

ρ

a uniform

equivalence since

be a

right

to the

left

The i n v e r s i o n

with

it

by

W

of

is

e

any

such

and s o

ρ

is

(p ) ^ = Ρ

.

equivalences and c l e a r l y

to the r i g h t uniform

between

G

the

left

uniform

structure.

with

the

same

structure.

on t h e o t h e r h a n d ,

with

both

g'

structure,

function,

Therefore

is

a uniform

structure

Some of

the

and

uniformly

in the e a r l i e r

are two-sided,

in the sense

that

has the p r o p e r t y

respect

left

structure

to the

uniform

if

G

then

G

t o the

r i g h t uniform s t r u c t u r e ,

versa.

if

is

For e x a m p l e ,

G

or t o t a l l y bounded w i t h r e s p e c t

with respect So f a r

t o the r i g h t uniform

being used.

to specify

The same i s

uniform

connected

structure,

are concerned,

which of

uniformly

to the l e f t

a l s o separated, uniformly

as these properties

unnecessary

separated,

sections

a l s o has

property with respect

is

is

continuous,

continuous,

uniform

t h e r i g h t uniform

G

ob-

the

i n v a r i a n t p r o p e r t i e s we h a v e c o n s i d e r e d

then

is

are

translation where

t r a n s l a t i o n s a r e uniform

is true with respect

G

topological

^

and r i g h t

equivalence

bounded.

a l s o a uniform

a neighbourhood

^ respect

totally

translation

g

left

is

left,

Although

is

i s uniformly

i s uniformly

totally

the entourages

(p^xp^) is

p^

G

translation.

denotes

and

is

G

Left

translation

we h a v e L·^ c

n e i g h b o u r h o o d of

to

since

to l e f t

G

t o t a l l y bounded on t h e

theory,

equivalence,

For i f

g e G

that if

uniform s t r u c t u r e .

so obvious,

equivalence.

that

G

sufficient

invariant with respect

element

course,

to the general

group with t h e

not quite

of

with the

and

vice

connected structure,

or t o t a l l y

and v i c e

therefore,

bounded

versa. it

is

t h e two u n i f o r m s t r u c t u r e s

t r u e of

any s e l f - i n v e r s e

subspace

is of

Topological groups a topological

group.

For what i s properties

of uniform

topological that

groups.

that

for

G

Ce ).

that F

for

t o be t o t a l l y a finite

The c o n d i t i o n

for

each

F

W of

Proposition of

of

group

G.

since

Η

for

F

as

F

left

respect

is

a filter

Cauchy i f

t o the l e f t

course,

H.K

F

subset

There

the group

H.K

of

corresponding

is

totally

e

in

G.

S.K

of

is

Then

totally

using

subset

a finite

bounded.

H.K,

subsets

bounded.

S unior

Therefore

and h e n c e

bounded,

group

G

as

asserted.

we d e s c r i b e

t h e Cauchy c o n d i t i o n

with

and we u s e t h e term when

G

are both l e f t

however,

as the

bounded

is

totally

structure,

filters

G

Μ

some f i n i t e

i s no d i f f e r e n c e

In g e n e r a l ,

is

g e n e r a t e d by t h e

Now

Τ

satisfies

uniform

distinguished

Consider

and s o i s

for

on t h e t o p o l o g i c a l

and c o n v e r g e n t

i n any c a s e . carefully

Then

bounded.

totally

Thus

Ν

W.Ξ = G.

a member

be t o t a l l y

is

some f i n i t e

Cauchy s i m i l a r l y . of

Κ

Η < Κ

φ : Η Κ Κ

G

hk

The c o n d i t i o n

for

Φ

t o be i n j e c t i v e

the condition

for

Φ

t o be s u r j e c t i v e

internal to

is given

by

conjugais

by

(heH,k£K). is

that is

that

the

be a normal

Η

=

the

Η η Κ =

(e);

H.K = G.

Topological groups When b o t h t h e s e c o n d i t i o n s and we r e g a r d and

K.

β : Κ

G

Note t h a t H\G

is

G

continuous,

the pair

open.

case

In t h a t

and a u n i f o r m

G

is

and

Η

K.

continuous.

isomorphism,

if

since G.

U

Κ Κ

consisting

of

The a c t i o n of

φ

K.

is

defined

are

acceptable

if

e m b e d d i n g when

is

Η

G/K

is (H,K)

is

topological

"'"^(αυ)

the equivalence

(e)

isomorphism

the pair

then

also

Urider

a topological

and h e n c e a

case.

Μ η Κ =

in addition.

that

and

i n any Φ

a : Η

II

as a t o p o -

i s continuous,

the semidirect

G

is

pairs

(x,0)

(x,y)

β

= U.K

w i l l be d i s -

Next l e t

pairs

(H,K)

isomorphic

(x,y)

(l,y)

quotient

condition

colleagues,

was t h o r o u g h l y

while

Κ

(I,χ

in this Η * Κ ,

subgroup is

the

χ = 1.

^y) .

It

case, moreover

spaces to right

was i n t r o d u c e d .

investigated

and we a d o p t t h e i r

the

Η

G/K.

who u s e d t h e t e r m " p r e s q u e i n v a r i a n t " the c o n d i t i o n

is

such t h a t

into

isomorphic to to

Η

y = 0

are a c c e p t a b l e ,

u s t u r n from l e f t

The f o l l o w i n g

i s p r o v i d e d by t h e

Here

such t h a t

transforms

topologically

topologically

product

considered e a r l i e r .

easy to see that

spaces.

Η Κ Κ

open in

normal subgroup c o n s i s t i n g of

is

and

action

open,

of

Η

bijection

s e n s e and t h e

assuming

is

and a

Η

later.

G =

and s o

of

H.K = G

is

The s t a t u s

An e x a m p l e o f

is

G/K

The i s o m o r p h i s m

«

isomorphism

Then

we o b t a i n

Still

an

group.

a uniform

therefore,

that

group

(H,K)

Φ

a c c e p t a b l e we f i n d

cussed

so t h a t

e q u i v a l e n c e when

these conditions,

IS o p e n i n

: Η

a topological

The homomorphism

Let us say t h a t

obviously

α

in the t o p o l o g i c a l

l o g i c a l group.

between

is

is

s e m i d i r e c t product of

an i s o m o r p h i s m

that

are subgroups

Η " Κ + Κ

(internal)

Φ

a r e g i v e n by t h e i n c l u s i o n s

Now s u p p o s e Κ

as the

are s a t i s f i e d

term

quotient

by P o n c e t

Subsequently,

[20^

horfever ,

by R o e l c k e [ 2 1 ] and

"neutral"

instead.

his

. . James Definition

(5.9).

is

in

neutral

there e x i s t s V.H c H.U

The s u b g r o u p

G

if

for

For

(5.10).

let

U

for

of

G

e

such

group in

G

G

that

may be s u b s t i t u t e d

Let

totally

condition,

Η

G.

Then

Η

b o u n d e d we h a v e of

is

g^W^ c V . g ^

.

S.W c V . S

By ( i v )

of

for of

Consider and

the

in

in

(5.1)

G .

G .

Then Since

of

e

intersection

Η

subset

there e x i s t s ,

W^

of

G.

some f i n i t e

a neighbourhood

normal

bounded s u b g r o u p

e

e

for

subgroups.

neutral in

of

Η c v.S

Η .

g^ £ S ,

V

Obviously

a s do open

be a t o t a l l y

some n e i g h b o u r h o o d

each element

in

for

G

such

W = W^^ π . . .

η W^ .

so

H.W c V . S . W • Β

B'

Β χ Τ

with projection

ρ ^(B')

restriction

a subspace

with

with the quotient

preimage ΧβIί

of

space

itself,

equicontinuously

space over

Thus

product

for

through uniform e q u i v a l e n c e s ,

the

notation.

the uniform Β

together

called

space over

space over

topological

regarded

sufficient

Also

X

ρ : X -»· Β,

as a uniform

can b e r e g a r d e d a s a u n i f o r m space

is

space

B.

function,

of

Β Thus

let

where

B'

is

a

u mjorm spaces over a oase iniform

space.

form s p a c e space of (b',x)

Then t o e a c h u n i f o r m

λ*Χ

over

the uniform such t h a t

g i v e n by

B'

p'(b',x)

to

B' If

of

X

X

is

= b'

.

A

is

Β

is of

B'

c Β

with

a

uni-

the subpairs p'

λ

with the r e s t r i c t i o n

is

the

in-

X^,

of

space over

Β

and

A

X

if

Note

fibrewise

Xj^

is

that if

Β

is

λ*Χ,

a subset

by

restricting

We d e s c r i b e

the c l o s u r e

A

dense in

is

of

fibrewise where

A

Aj^

for

d e n s e in

λ : Β'

Β

X is

above. {Xj }

fibrewise

the uniform

with

is

uniform

coordinates

a family

of uniform

product

Π^Χ^

product

lie

JIX^

spaces over

i s defined

consisting

i n t h e Scime f i b r e .

of

Β

t o be the points

Thus

the

Π^Χ^

subspace

all

of

comes

whose

equipped

projections ..

: Π3Χ.

X.

which are u n i f o r m l y

continuous

the pull-back

of

λ*Χ

respect to a uniformly identified

with

space over

Β

B'

where

special

base which are important

over X

is

B.

over

function B'

Note Β

λ : B'

that

with Β

can be

r e g a r d e d a s a uniform

λ. classes

of uniform

i n our t h e o r y .

i s uniformly

which the p r o j e c t i o n In [ l l J

space

continuous

with projection

which the p r o j e c t i o n

functions

a uniform

>4„ X,

There are s e v e r a l

1.

consisting

and t h e p r o j e c t i o n .

dense^ i n b e B.

λ*Α

If

for

λ*X

as a uniform space o v e r

structure

each point

of

where

over

and t h e p r o j e c t i o n

In c a s e λ*Χ

a uniform

we r e g a r d

as fibrewise

as

χ X

X

.

the uniform

then

B'

= p(x)

c l u s i o n we may i d e n t i f y X

is defined,

product

\(b')

space

open.

One i s Another

i s a l o c a l uniform

d e n s e a l w a y s means f i b r e w i s e

spaces over a the c l a s s i s the

equivalence.

dense.

for

class Note

I.M.James that

if

the uniform

space

so does the pull-back Β

and u n i f o m i l y

Proposition

D

of

X

p"^(E[b])

is

ζ

B'

X

open i f

C

all

c ρΟ[ς]

for

all

to

loss

Then i f

and s o

and t h e n

ξ £ DtxJ

for

over a base.

χ

£ D[Xg]

.

so that

over

X

have the o r i g i n a l

.

choose

that

So i f

β £ E[b]

η £ 0[ζ]

χ χ

.

Then

as

property

property

if

in Section

Definition

(7.2). all

Β Also B.

is

for

1.

For

The u n i f o r m the fibres

of

for

is

that

seme ζ £ X^ ,

an o b v i o u s way spaces

t h e uniform

space

the fibres

of

procedure i s

Section

4 is

invariant

too

concerned properties

example space

X

X

separated.

always separated, Β X Τ

entourages

of uniform

this

the uniformly

the

for

is

all

Although

satisfied.

required.

there

we may s a y t h a t

is

we h a v e

£ Οίζ]

w h e r e t h e Cauchy t h e o r y o f satisfactory

e X^

,

spaces

fibrewise

property.

considered

sDace o v e r

that

so

condition

and

Hence

has the f i b r e w i s e

simplistic

itself.

D ,

seme

β £ pD[xJ

Specifically

X

Thus

entourage

Then we can

we may a s s u m e t h a t

β £ E[b]

a corresponding

if

B.

for each such

open.

the stated

Given a p r o p e r t y of uniform

is quite

Β

ξ e Xg

b = ρ(η)

of g e n e r a l i t y

b £ Εϋβ]

separated

of

symmetric and

suppose that

are symmetric.

it

We p r o v e

space over if

space

required.

Conversely,

Β

Β .

and o n l y Ε

then

e v e r y uniform

λ : Β'

i s uniformly

β £ Β

and s o

as

Without r e a l

ρ

for

property

b c Β .

in relation

b £ Ε[β]

£ 0[η],,

that

,

be a u n i f o r m

uniformly

for

has e i t h e r

function

an e n t o u r a g e

Ε ,

to define

Β

there e x i s t s

symmetric

then

Let

over over

continuous

For s u p p o s e

Ε[β]

λ*X

(7.1).

The p r o j e c t i o n

X

fibrewise

each separated

are

over

Β

as a uniform separated, uniform

is

space

fibrewise

over

as a uniform

space

T.

We p r o v e

u mjorm spaces over a oase Proposition

(7.3).

Suppose t h a t Then

X

Β

is

R

is

R

Since

X

is

X XgX

is

just

of

B. is

Since the just

the diagonal as

ΔΧ

space over fibrewise

Β .

separated.

t h e e n t o u r a g e s of the i n t e r s e c t i o n

latter

reduces

to the

product

hence

R

of diagonal

X x„X . ΰ

t h e i n t e r s e c t i o n of

ΛΧ,

X.

itself

R

with

reduces

to

required.

is closed

t h e s i t u a t i o n when Proposition

of

the f i b r e w i s e

separated

It is easy to see

uniform

is

i n t h e p r e i m a g e of

fibrewise

the diagonal,

uniformly

X

be t h e i n t e r s e c t i o n

the preimage

only if

be a uniform

s e p a r a t e d and

contained

the entourages ΔΒ,

is

X

separated.

For l e t Then

Let

101

that in

ΔΧ

(7.4).

X X x^X

is

Let

if

fibrewise

.

separated

Our n e x t r e s u l t

if

and

concerns

open. X

be a uniform

open p r o j e c t i o n .

equivalence

is

space over

Then t h e p r o j e c t i o n

and o n l y i f

the diagonal

D

is a

with

local

function

Δ : X Η- X XgX i s uniformly

open.

For s u p p o s e X

admits a base of

tne p r o j e c t i o n (D[x]

for

all

X

χ e X Δ

required.

If η

X £ X ,

D

is

(X XjjX)

equivalence.

Then

Δ

a basic

entourage

then

C ADCXJ

i s uniformly

i s uniformly

there e x i s t s

all

i s a l o c a l uniform

symmetric entourages which are transverse

and s o

(E[x] X E[x]) for

ρ

ρ .

X D[x])

suppose that of

that

open.

open.

If

a symmetric e n t o u r a g e

D

Conversely i s any

Ε

of

X

entourage such

η (X ΧβΧ) c Δ Ο [ χ ] and t h e n

Ε

is

transversa to

ρ ,

as

that

to

102

. James

Definition

(7.5).

wise t o t a l l y Thus

Β

The u n i f o r m

bounded i f is

always

space over i t s e l f .

space over

Again i f

is

formly wise

fibrewise

Proposition

X

over

of

X

totally

Β χ Τ B,

is

bounded, (7.6).

on a u n i f o r m

as a uniform Let

X

uniformly

open p r o j e c t i o n .

and t h a t

X

is

fibrewise

space

bounded.

as a u n i f o r m

totally

bounded,

X

then

X

X/G .

Β

bounded.

is

uni-

is

fibre-

We p r o v e

space over

that

T.

group a c t i n g

over

be a u n i f o r m

totally

fibre-

t o t a l l y bounded s p a c e

space

Suppose

is

bounded,

fibrewise

for each

Β

are t o t a l l y

a t o t a l l y bounded t o p o l o g i c a l

equicontinuously

totally

the f i b r e s

Also

as a uniform G

all

space

B,

with

totally

bounded

Then

X

is

totally

and l e t

Ε

be a s y m m e t r i c

for

b ε Β

and

subset

S

bounded. For l e t

D

b e any e n t o u r a g e o f

Β

such that

entourage

of

Since

Β

is

of

X

such t h a t

finite

number o f

totally

each fibre

is

subset

of

then

Τ

and s o Thus

fibres.

totally X

p(s) (5,t)

bounded,

for

some

= ρ(ς)

for

for

D c D[T] = X ,

(2.15),

some

uniformly

(7.7).

if

some

and s o

t ε T,

the union of

exists

.

ξ ε D[x],

a special

all

is

t o t a l l y bounded,

p~^pS c D [ T ]

The u n i f o r m

connected

ρ ^pS

s ε S,

all

a finite

and s o t h e r e

which proves

is essentially

Definition

Now

The u n i o n i s

such that

ε D

- pDLxJ

bounded t h e r e e x i s t s

Β = E[pS].

b e E[p(s)]

therefore

E[bJ

X,

ρ(s)

a

since

finite

So i f

x e

ε E[b] c pD[x],

hence

therefore (7.6).

a

χ ε Χ^^.

ξ ε ρ

^p(s)

χ ε D » D[TJ

Our e a r l i e r

.

result,

case. space

the fibres

X of

over X

are

Β

is

fibrewise

unifonnly

connected. Thus

Β

is

always fibrewise

self.

Also

Β >· Τ

uniform

space over

is B,

fibrewise for

uniformly

connected over

uniformly

connected,

each uniformly

connected

it-

as a

uniform

orm spaces over a ase space

Τ .

Again i f

G

group a c t i n g uniformly then

X

is

over

X/G .

fibrewise

is

equicontinuously uniformly

(7.8).

Let

X

uniformly

open p r o j e c t i o n .

connected

and t h a t

i s uniformly For l e t

X

is

on a u n i f o r m

connected,

be a uniform Suppose

space

as a uniform

fibrewise

space over

that

Β

is

uniformly

D

X

space

3,

with

uniformly

connected.

be any s y m m e t r i c e n t o u r a g e o f

b e B.

of

Let

Β

ζ,η

such

that

e X .

Then

connected.

chain to a point

ρ(ξ)

e Β ,

I assert

x^ £ Xj^

,

that

where

ζ

X,

and l e t

ECb] ^ pOLXj^J

Then

can be j o i n e d by an E - c h a i n uniformly

topological

connected.

be a s y m m e t r i c e n t o u r a g e each point

connected

We p r o v e

Proposition

X

a uniformly

and

since

Ε

for

ρ(η) Β

is

can be j o i n e d by a D-

i = l,...,n

.

This

is

i trivial true

for

for

some

(Xj^jX^^j^) ξ

1 = 1 ;

e D

make t h e

i < η . for

inductive

Then

hypothesis

£ E [ b ^ ] c poLxj^J

some

^ ^b.

'

n' X,

it

and

^^ r e q u i r e d .

can b e j o i n e d by a D - c h a i n t o some p o i n t

then since

that

η' £

is

so Thus

b

,

η

and

c a n be j o i n e d t o η by a D - c n a i n ( i n Xj^ i s uniformly connected. Thus ζ can be j o i n e d

to

η η

by a D - c h a i n a s r e q u i r e d .

(2.17),

is

essentially

a special

We s e e , t l i e r e f o r e , t h a t over a base properties point.

c a n be d e f i n e d of

This

uniform

t h e t h e o r y by

proposition uniformly

(7.9).

result,

case.

fibrewise

properties

which reduce

to the

of

uniform

a s we s h a l l

however, see.

is

spaces

corresponding

s p a c e s when t h e b a s e s p a c e r e d u c e s

simple procedure,

t h e Cauchy t h e o r y , of

Our e a r l i e r

inadequate

We b e g i n our

to a

for

discussion

proving Let

open p r o j e c t i o n .

X

be a u n i f o r m Let

F

space over

B,

be a Cauchy f i l t e r

with on

X

I.M. James such t h a t

p^F

member o f that

converges

trace

For

let

of

G

D

on

for

converges

we h a v e

to

b

and t h e n

Xj^ .

Since

G

runs

through

the

sufficiently

is

such

if

for

itself

X . Ε

By

of

Xj^ ,

for

so

F,

we

have

Since

p^F

sufficiently Thus

by t h e s e t s X

(7.1)

Β .

c D[Xj^] .

of

meets

D"^CM]

D ^[M],

and

Μ

runs

this

proves

small meets

where

through

0

the

the r e s u l t .

i s m i n i m a l Cauchy t h e n t h e

The u n i f o r m

each point converges

that

space

b ε Β to

b

this

reduces

X

over

In

trace

Β

is

of

F

fibrewise

and C a u c h y f i l t e r

,

F

itself

to completeness

Β

is

a point-space.

complete,

as

a uniform

fibrewise

complete

F

converges

on

to

X

some

uniform

in

Moreover

space spaces

over

the

Β

ordinary

is

itself.

always

fibre-

Pull-baclcs

over a base

are

also

of

fibre-

complete. We h a v e

at

once

Proposition

(7.11).

Let

space

Β .

Β

over When

uniform we

F

X ε Xj^ .

s e n s e when

wise

F

p,F

Note

wise

entourages

(7.10)·

that

point

if

of

pM c ECb]

generated

of

every

defined.

some e n t o u r a g e

Μ c p"^(E[b]) is

G

Then

defined,

Definition complete

is

s m a l l members o f

particular, X. b

Xj^

b e any e n t o u r a g e

ρ ^ E [ b ] • Y

over

To d e f i n e procedure

ψ ,

as for

separated,

entourages

complete

over

is

uni-

χ ε X . group

space

Then

X.

acting X

X/G .

X

be a u n i f o r m

d e n s e s u b s e t of

space over

X .

such that

continuous

Y

Β

Y

and

be a

space over function

Β

over

continuous

B.

function

ψ|Α = φ .

as a f u n c t i o n , the ordinary

Let

complete uniform

be a u n i f o r m l y

S i n c e we a r e o n l y a s s u m i n g

the

topological

o n e and o n l y o n e u n i f o r m l y Β

of

each point

on t h e u n i f o r m

separated fibrewise φ : A

complete.

the projection

Corollary

(7.15).

Further-

space.

c o m p l e t e for

Proposition

Τ .

with projection a

fibrewise

is

complete over

proves

fibrewise

space Β

point

Hence

which is

be the i n t e r s e c t i o n

R[x]

fibrewise

,

fibrewise

also

is

χ

th

point

an a d h e r e n c e

Β χ Τ

space over

then

R

t o some

point since

that

of

The

is

converges to

filter Xj^

(7.9).

χ

as we h a v e s e e n i n S e c t i o n

open,

by

on

Then

uniform

the separated quotient For,

The t r a c e

e v e r y c o m p l e t e uniform

Let

space

be a Cauchy

and s o c o n v e r g e s

also

equivalence

the uniform ,

is defined,

of

b .

is complete.

over

l o c a l uniform

of

G

In p a r t i c u l a r we s e e

more,

Corollary

F

F

to

and s o a l i m i t

the refinement

and l e t

converges

a l s o a Cauchy f i l t e r

X e Xj^ , of

Β

vje u s e e s s e n t i a l l y

e x t e n s i o n theorem t o be f i b r e w i s e

we o n l y h a v e u n i q u e n e s s

of

limits

t h e same

(4.21)

above.

separated,

not

for convergent

filter

106

ι.Μ. James

in a restricted q^H

sense.

converges

converge that

thus

defined

be any e n t o u r a g e

D

that

of

assert

such

Since

on D

A is

Μ X Ν C D

open

and s o

ilarly

φΝ c Γ [ ψ ( η ) J

we h a v e

that

all

D

into of

X)

the

cannot

Yj^ .

To

on

Then

on

Ε

entourage entourage

(• X„ υ

can

function

and t h e n e x t e n d e d

is

is

t o a uniformly

is in

f i b r e w i s e d e n s e in X„ and (λ*Χ),'^, ij , ij

complete.

So we can u s e

to construct

the canonical

of

i

the

identity

that

a uniform

separated

to

ijj'j

and o b t a i n

ψ

fibrewise

ψ

.

Now

and

a uniformly

function ψ'ψ

on

ψ'

equivalence

fibrewise

the

λ*(1Χ)

.

is

B'

.

λ *X , identity We

are mutually

over

completion

is

for

inverse,

Thus

natural

the

with

pull-backs.

In t h e c o u r s e o f

this

section,

observed,

we h a v e made c o m p a r a t i v e l y

structure

of

possible

ρμ = λ ρ '

function

continuous

and f i b r e w i s e

the pull-back

(7.15)

respect

Β

(X*X)g,

i|j'jX*X

apply

continuous,

function

:

B'

where

respect

X*Xg

separated

continuous

over

with

space over

such t h a t

the canonical

Β'

the

φ.

a uniform

i n t h e same way a s b e f o r e ,

Ψ'

of

as

function

fibrewise

(7.15),

φ^

be u n i f o r m l y

over B' . On t h e o t h e r hand λ*(1Χ) λ * ( Χ „ϋ) , since iX i s f i b r e w i s e dense is

to

comes e q u i p p e d w i t h a uniformly

be p u l l e d back t o a u n i f o r m l y φ : λ*Χ -»• λ*Χ,, D

We r e f e r

t h e q u e s t i o n of n a t u r a l i t y

Let

the pull-back

1 uy

the base space.

to develop a similar

a s t h e r e a d e r may h a v e little

This s u g g e s t s theory for

u s e of that

uniform

the

uniform

i t m i g h t be spaces

over

1 ιυ

I.m.jamex

a topological

base

For a c o m p l e t e l y

space,

satisfactory

to proceed to a further are

(in a c e r t a i n

vertically. studied

sense)

These are

stage

has b e e n done i n

theory,

however,

it

is

the fibrewise III

of

horizontally

mentioned.

necessary

uniform

spaces

[11 1.

The r e s u l t s

The work o f Hunt [ 6 ] on u n i f o r m

which

and u n i f o r m

s e c t i o n may s e r v e a s an i n t r o d u c t i o n

general r e s u l t s .

L 8 J.

and c o n s i d e r h y b r i d o b j e c t s

topological

i n [ 9 ] and i n C h a p t e r

in the p r e s e n t

i l s o be

and t h i s

I have given

to these spreads

more

should

8.

Uniform covering spaces

I a s s u m e t h e r e a d e r t o be f a m i l i a r t h e o r y of c o v e r i n g Godbillon [5]

or

spaces in the t o p o l o g i c a l

fully that

of

covering

group, which

theory

there

of

o f c o v e r i n g map.

In t h e uniform

we n e e d t o s t r e n g t h e n lence in a similar

stronger condition

for 1.

R

[22].

This

section

suggests

equivalence

I give

and

l o c a l uniform

that find

equiva-

and c o m p a t i b l e . still.

r e l a t i o n or

2 we h a v e d e f i n e d

the

the

terms

What we now n e e d i s a

Let us say t h a t t o the uniform

R

is

structure,

strongly if

there

the

case,

that

R » D = D » R

each symmetric b a s i c entourage

D .

The a p p r o a c h a d o p t e d by L u b k i n [ 1 5 ]

This i s is

an

difference

t h e o r y we c a n e x p e c t t c of

of

.

an e s s e n t i a l

b e an e q u i v a l e n c e

In S e c t i o n

with respect

a base such

is

[16]

d e a l t w i t h more

the f i n a l

topological

the d e f i n i t i o n

let

X .

weakly compatible

(8.1)

is

fashion.

To s t a r t w i t h , uniform space

local

in

i n Massey

t h e form w h i c h s u c h a t h e o r y m i g h t t a k e ^

between the notion

exists

as

t o d e v e l o p a uniform v e r s i o n

s p a c e and i n

In the t o p o l o g i c a l

compatible,

sense,

t h e c a s e where t h e

[ 3 ] but see a l s o Taylor

ought t o be p o s s i b l e

the notion of outline

the theory concerns

a topological

in Chevalley it

classical

( w i t h some t e c h n i c a l d i f f e r e n c e s )

An i m p o r t a n t b r a n c h o f base space i s

with the

unrelated.

112

ι.Μ. James

for

example,

action

of

if

Β

over

alence

Β

discrete

Proposition

is

just

Β

Now

and l e t

D ·

D

pD[x]

is

X £ X .

In f a c t

entourage

of

injective

(ξ,η)

Β

if

D[x]

open.

s

:

ECb]

a uniform

X

(6.6)

strong

D[x]

onto ρ

to the p a r t i a l

(X

e χ

)

each

on

equithe

and our o b s e r v a t i o n

that

t o show t h a t is

a covering

of

X

p{x), Ε =

where ρ .

E[p(x)],

maps

for

uniformly

be a uniform

of

to

map.

X/G.

• Β of

X

X ,

space over

X

let

φ:Χ-«·Υ,

be t h e p r o j e c t i o n s .

which s a t i s f i e s

(φχφ)Ο

is

(8.1)

an e n t o u r a g e o f

respect to

S = (qxq)~^A,

For e x a m p l e l e t equicontinuously the uniform

space

see that

X/H

subgroup

Η

is of

q:Y->B

G

with respect Y

D

which s a t i s f i e s

R = (pxp) (8.1)

then

with

uniformly

properly discontinuously

By c o m b i n i n g

^Δ

transversality.

be a d i s c r e t e g r o u p a c t i n g

on

t h e l a s t two r e s u l t s we

a u n i f o r m c o v e r i n g s p a c e of G .

i s an e n t o u r a g e

to

and s i m i l a r l y w i t h

and u n i f o r m l y X .

If

and

X/G

for

each

I.M.James For t h e r e m a i n d e r

of

this

s e c t i o n we work a l m o s t

i n t h e c a t e g o r y of u n i f o r m s p a c e s o v e r a uniform If

X

is

a uniform

a uniform

covering

space over s p a c e of

unifotm

self-equivalences

uniform

covering

which

t h e y form

(8.2).

(8.4).

and

g^

agree at

φ = g^

connected.

Let

X

Β .

D

transformation

properly

X

i s uniformly

as

required.

ρ .

Let

such t h a t

X ,

: X -«• X

Let

i s uniformly

X

£ D

for

since

are uniformly

be a u n i f o r m

locally

by

such

connected.

that

moreover X ,

G ,

by

(2.7) ,

in t h i s

connected

case,

uniform

the uniform

Poincar^

transverse

be a uniform

c o n n e c t e d we o b t a i n from

(8.5).

,

g £ G

which i s

X

by t r a n s v e r s a l i t y ,

φ , id

X/G

discontinuous.

φ : X

(φ (χ) , χ )

dis-

.

be a u n i f o r m l y

properly

the

transformation,

throughout

Therefore

b e an e n t o u r a g e o f

= X ,

the functions

X

χ

Then t h e a c t i o n of

i s uniformly

to the p r o j e c t i o n

Β

and u n i f o r m l y

the uniform Poincare group

For l e t

where

Suppose t h a t

and t h e n c h o o s e an e l e m e n t

Consequently

s p a c e of

Proposition

X .

covering

precisely

φ(χ)

space

X .

self-equivalences

a uniform

uniformly

Then

be a g r o u p o f u n i f o r m

the

group

P o i n c a r e g r o u p of

is

φ

is

called

and t h e d i s c r e t e

X

is

group of

X

φ : X

Then

X

These are

c o v e r i n g s p a c e of

since

covering

of

Β .

a uniform

·

Proposition

over

equicontinuous

πφ = TT ^ i r g ^

is

X ,

connected uniform

χ e X

B.

we s h a l l n e e d t o c o n s i d e r

is

.

X

if

X

If

- x.g

and p a r t i c u l a r l y

t h e uniform

G

base space

so that

choose a point φ(x)

let

uniformly

continuous,

of

i s called

the uniformly

action is

Β ,

transformation s

For e x a m p l e , of

Β ,

entirely

covering

some p o i n t ρφ = ρ . continuous

(2.7)

that

χ £ X . Since and

since

φ = id

c o v e r i n g space of Then t h e a c t i o n

,

Β , of

unijorrn

the uniform

and

is

E'

c Ε

is

all

transverse

uniform c o v e r i n g

b = p(x)

at

(pxp)D

b ε Β ,

to ,

ρ .

where

D

I assert

then

equicontinuous.

be as b e f o r e

φ : X ->• X .

.

sections

where

satisfies

that

(φ(ς),φ(χ))

transformation

The p a r t i a l

the point

connected,

b ,

and s o

and

if

ε D

(8,1)

(ξ,χ) for

ε D',

each

For c o n s i d e r

(3.12)

of

and s o

ξ ε D[x] = s^E[b]

ρ(ς)

ε E'tbJ,

φ(ς)

ε φs'^E'[b]

(φ(ς),φ(χ)) In t h i s

topology.

thus

ε D

also

=

by t h e

(ρ(ξ),ρ(χ)) .

Now

the

E'[b]

by

ε Ε'

in the topological

and

and

so

φ ,

not

t h e uniform

if

Φ

sense

then Thus i t

is

we h a v e o n l y

continuity.

a covering

Φ is

In

base

covering

a covering

i s unnecessary

spaces

over

used fact

transformation transformation to

distinguish

t h e u n i f o r m P o i n c a r e ' g r o u p and t h e t o p o l o g i c a l

connected

so

Therefore

i t may b e o b s e r v e d ,

sense.

uniform

hypothesis,

as r e q u i r e d .

of

group for

is

coincidence

ζ ε D'[x],

C 0[φ(χ,],

t h e argument shows t h a t

i n the uniform

We h a v e

ς ε s^E'Cb]

argument,

the continuity

ρφΞ' = p s ' , . X Φ IX;

φs^E·Γb] = s^^^jE'Cb],

theorem

between

i s uniformly

sections

where

agree

X

Ε =

D' = D η ( p x p ) ~ ^ E '

partial

j ^

and

c o n n e c t e d for

D « D

where

spaces

Poincare' group on

For l e t E'Lb]

tovenng

a uniformly

Poincare

locally

space.

By c o m b i n i n g

Corollary

(8.6).

covering

s p a c e of

the

last

Let Β ,

X

two r e s u l t s w i t h

be a u n i f o r m l y

where

Β

(8.2)

we

connected

i s uniformly

obtain

uniform

locally

I.M. James connected. form

Then t h e n a t u r a l

covering,

where

G

Among t h e u n i f o r m special

role

following

the

uniform

c o v e r i n g s of

p l a y e d by t h o s e

ττ : X

X/G

is

a uni-

Poincare group of

a given base space

which are regular

in

X .

a

the

sense.

Definition regular

is

is

projection

(8.7).

if,

The u n i f o r m

first,

t h e uniform

X

covering

is uniformly

P o i n o a r ^ group of

X

space

connected

acts

X

of

and,

Β

is

secondly,

transitively

on

the

fibres. For e x a m p l e , of

the uniformly

neutral

of

X/G

Proposition Β .

over

Β .

X e ρ ^ (b)

X

is

Let

this,

choose

ε ρ ^(b)

is

where

be a u n i f o r m l y a uniform a point

exists

b e Β is

φ

Therefore

φ = ψ

Χ ,

Let X

G

be a

through uniform

throughout

at

(discrete)

equivalences.

the diagonal

X X Τ .

If

t h e a c t i o n on

uniformly

properly discontinuous (8.2),the

I T : X X T - > X X q T

action

is

uniform

by

.

X

natural

is

space

function

point

(2.7),

Since

transformation

Then

ρφ = ρ = ρψ .

since

Χ

is

(8.8). on t h e u n i f o r m

Consider,

action

covering

transformation.

and a

group a c t i n g

Τ ,

by

then the

covering χ

This proves

G-space

and h e n c e ,

the

the projection.

a uniform

which agrees with

connected.

If

continuous

covering

ψ : X -»• X

uniformly

X .

be a r e g u l a r u n i f o r m

ρ : X -· X φ

,

be a g r o u p of u n i f o r m

properly discontinuous

Then

To s e e

φ(χ)

by

(6.8).

Let

G

connected uniform

and u n i f o r m l y

covering

of

let

of

G

any

discrete

on t h e u n i f o r m

uniformly then

for

so i s

projection

space

product

equicontinuous

and

the a c t i o n

X χ Τ

on

mjorm covering spaces is

a uniform

X Xq Τ ,

covering.

in the usual

Proposition space

of

let

G

H e r e we h a v e w r i t t e n

(8.9).

Β ,

way. Let

where

(X''T)/G . a s

X

Β

be t h e u n i f o r m

be a r e g u l a r u n i f o r m

i s uniformly

locally

P o i n c a r e group of

covering

connected,

X .

Then

and

the

projection ρ : X is

Τ Η- X/G = Β

a uniform Here

below,

covering,

ρ

is defined

where

natural

for

ρ

is

i·'

the f i r s t

>

projection

and

π

denotes

shown the

space.

X

π

π Χχ^Τ

> X/G

G

ρ

ρ

is

a uniform

are the p r o j e c t i o n s is

Τ .

through the commutative diagriim

p r o j e c t i o n onto the o r b i t

XxT

Now

each d i s c r e t e G-space

a l s o a uniform In f a c t ,

t u t e s a functor

covering, π , by

since

(8.2).

By

is discrete, (8.3)

and s o

therefore,

P

covering.

u n d e r t h e c o n d i t i o n s of from

(8.9),

X x_ G

consti-

t h e c a t e g o r y of d i s c r e t e G - s p a c e s and G-

maps t o t h e c a t e g o r y o f u n i f o r m uniformly

Τ

continuous functions

r e s u l t s w h i c h may h e l p t o

covering over

Β .

illustrate

spaces over

Β

and

We p r o v e t w o f u r t h e r

the behaviour of

this

functor . Proposition s p a c e of Τ

(8.10).

Β ,

where

Let Β

be a d i s c r e t e G - s p a c e ,

g r o u p of onlv

if

X . G

Then

X

be a r e g u l a r u n i f o r m

i s uniformly where

X x^ Τ

acts transitively

G

is

locally

on

Τ .

connected.

the uniform

i s uniformly

covering

connected

Let

Poincar^ if

and

110

I.M. James For

T/G

suppose

that

is uniformly

uniformly given

X x^ Τ

connected,

continuous

there

exists a

orbit.

Thus

Since

X

: Χ x^ Τ

if

t,t'

T/G

t Τ

is a

are

η ,

,

where

t ^^ , t 2 , . . . , t ^

In o t h e r words t h e a c t i o n

Conversely

t

prj/G

Then

chain = f

seme i n t e g e r

since

surjection.

^ = for

i s uniformly connected.

suppose t h a t

i s uniformly

G

is

transitivity

connected,

SO

Τ

implies

in the

so i s

that

same

transitive.

acts transitively

t Τ and h e n c e s o 13 t h e p r o j e c t i o n

However,

lie

Χ x [t},

ΤΓ{Χ χ { t } ) π(X χ { t } )

on

Τ .

for

each

point

C χ x^ Τ .

= X χ _ Τ , and LI

X

i s uniformly

Proposition s p a c e of X/H for

is

(8.11J.

Β ,

every

where

Β

to

subgroup

To s e e k +

X

Let

where

equivalent

this, X

k(x)

continuous

X

connected. X

i s uniformly

of

m -

= (x,e)

,

locally

the

covering Then

space over

Poincare group

the uniformly

proof.

connected.

as a u n i f o r m

t h e uniform

consider G

be a r e g u l a r u n i f o r m

X x^, G/H

Η

This completes

continuous

G

Β ,

of

X .

functions

X , and

m(x,g)

= x.g

.

These induce

uniformly

functions

X/H -- Β

in the t o p o l o g i c a l

P o i n c a r ^ g r o u p of

One may c o n c l u d e ,

conditions

ρ

and t h e r e f o r e

suppose that

Β , X

Since

and s o

a uniform c o v e r i n g

a uniform

More g e n e r a l l y , covering

is

b € Β , and i f

can be r e p r e s e n t e d

ο + λ = λ

ρ ,

pathwise-conriected

each p o i n t

(α+λ,λ)

ρ : Β/π^ (Β) ->• Β .

and u n i f o r m l y

covering

E[bJ

Now t h e p r o j e c t i o n

bijection

Β .

121

space

of

reasonable sense is a

,

Appendix:

filers

In t o p o l o g y nowadays

the v a l u e of

is generally

recognized.

the concept,

i n t h e m a i n p a r t of

an a p p e n d i x

a brief

no d o u b t i t

will

Definition

a c c o u n t of

A filter

F

of

the t e x t ,

the n e c e s s a r y to

non-empty s u b s e t s

theory,

many o f

on a g i v e n

set

of

X

X

a member of

F

(ii)

the i n t e r s e c t i o n

of

subfamily

In a s e t

Such f i l t e r s family for

the most immediately obvious

of

all

s u p e r s e t s of a g i v e n

have t h e p r o p e r t y that

o f members i s

finite

s e t s but i n f i n i t e

s e t form

a l l members o f ment of

a finite

It

is often

family

a filter

F^

F,

F

is

a

non-empty

This

sets contain

is

F^

i s empty.

such t h a t

those

subset. of

always

filters

are

any

the

case

which do n o t

subsets

of

an

the i n t e r s e c t i o n

(A c o f i n i t e

subset

is

of

the

ccmple-

subset). convenient to specify

o b t a i n e d by t a l c i n g

follows .

that

filters

For e x a m p l e t h e c o f i n i t e

of g e n e r a t o r s ,

For t h i s

a non-

the i n t e r s e c t i o n

a l s o a member.

have t h i s p r o p e r t y . infinite

readers. ~

a member o f of

as

F. X

which c o n s i s t

is

of

although

my

such

each s u p e r s e t of

a finite

is

use

I am i n c l u d i n g

(i)

member o f

filter

I h a v e made e s s e n t i a l

a l r e a d y be f a m i l i a r

(Λ.1).

empty f a m i l y

Since

t h e c o n c e p t of

i.e.

all

t o wor)c t h e

a family

a filter

such t h a t

s u p e r s e t s of members of

family

has t o

sati sfy

by d e f i n i n g

the the

filter

a

is

family.

one c o n d i t i o n ,

as

Appendix: filers Definition

(Λ.2).

empty f a m i l y s e c t i o n of

Β

A filter

b a s e on a g i v e n

of n o n - e m p t y s u b s e t s

each f i n i t e

subfamily

Of c o u r s e e v e r y f i l t e r

of

of

B

set

X

X

s non-

such t h a t

contains

c a n be r e g a r d e d

is

tne

a member o f

as i t s

the

filter

of

example,

consider

of

its

a sequence

The e l e m e n t a r y f i l t e r

by t h e f a m i l y

Note t h a t

it

to define

the

the

φ : X

If

defined

by t a k i n g

In c a s e

F

X

is

is

φ a s

In g e n e r a l X

in the

sequence

{Xj^

,...} than the

set

is

X.

the

for

set

filter

k =

ix,^]

we h a v e t o know t h e o r d e r o f

Y

the

on

X.

a subset

φ

Y of

1,2,.. in

ordei

the terms

If,

then a f i l t e r t h e members of

and F

φ

to

however,

in

b a s e on

is

and

φ^F F

Y on

X

are Y

is

as a b a s e .

t h e i n c l u s i o n we r e f e r Y .

G

φ

for

is

G

which the

surjective,

are s a t i s f i e d

φ*0 .

i n c l u s i o n we r e f e r it

X

In c a s e to

φ*0

only defined

X

Y.

non-

then

and t h e is

on

preimages

t h e preimages are a l l

be t h e c a s e when

Of c o u r s e ,

where

hand t h a t we h a v e a f i l t e r

i s d e n o t e d by

the

the filter

a subset

as the trace

cf

of ϋ

when e v e r y member o f

X.

Definition is

of

on t h e o t h e r

as w i l l

meets

X

t h e r e may be members o f

so generated and

on

Images of

c o n d i t i o n s for a f i l t e r

γ

be a f u n c t i o n ,

a filter

are empty.

empty,

F

with the

t o know m o r e

the e x t e n s i o n

Suppose,

G

points

sequence.

sets.

in

of

subsets

necessary

filter;

Now l e t

to

of

For another t y p e

^^^^^

associated

generated

is

supersets.

Β .

own b a s e .

A l s o e a c h n o n - e m p t y s u b s e t c a n be t a k e n a s t h e b a s e o f consisting

inter-

(A.3).

a filter

a member of In t h i s

F'

Let on

F X

be a f i l t e r

on

X.

A refinement

s u c h t h a t e a c h member of

F

is

of also

F'. s i t u a t i o n we s a y t h a t

F'

refines

F,

or t h a t

F

I.. James is

refined

F".

by

When t h e p o s s i b i l i t y

b e e x c l u d e d we d e s c r i b e For

example,

of

let

F

e a c h member o f

and

base

G.

F

Μ η N,

No common r e f i n e m e n t F

and

G

and

filter Y

Y

are

sets.

F

on

X.

such t h a t

Refinement filters

Then

φ*G

φ*φ*Ρ

φ^φ*6

is

is

such

to

Ν e G,

of

that

family

constitutes

which r e f i n e s course,

F

both

if

there

meet.

φ : X ->· Y

be a f u n c t i o n ,

refined

refines

G

by

F,

where

for

each

for each f i l t e r

G

defined.

imposes a p a r t i a l

on a g i v e n

is called

which do n o t

X

Then t h e

and

can e x i s t ,

let

Also

G.

a filter,

is

strict.

be f i l t e r s on

Μ ε F

where

For a n o t h e r e x a m p l e ,

on

G

and s o g e n e r a t e s

a r e members o f

X

and

as

m e e t s e a c h member o f

intersections

a filter

the refinement

F = F'

that

set.

o r d e r on t h e c o l l e c t i o n

A m a x i m a l e l e m e n t of

an u l t r a f i l t e r .

The f o l l o w i n g

the

of

collection

criterion

is

often

useful . Proposition

(A.4).

ultrafilter

the following

if

Μ υ Ν £ F,

For a f i l t e r

where

subsets

N'

Now

is

F'

since

Ν

of

F.

Μ € F'

condition since

F'

of

Therefore

but

F

Μ /

F

then

is

Μ /

Μ u N'

e F

a refinement

of

filter,

since

This ccmpletes

of

F

Μ e F and

n o t an

F

admits a s t r i c t

or

But then

F.

Ν /

is

of

F. The

F'.

strict,

refinement and s o

X - Μ ε

which i s contrary to

Μ ε F'

Ν ε

ultrafilter.

Μ υ (X-M) = X £ F

F,

sufficient:

form a f i l t e r

is

the proof

n e c e s s i t y r e s u l t we o b t a i n

t o be an

and t h e r e f i n e m e n t

X - Μ ε F.

is

X

n e c e s s a r y and

but

F

suppose t h a t

implies that

definition X-M.

such that

on a s e t

then e i t h e r

Μ υ Ν e F

a refinement

Conversely If

X

condition

Μ,Ν c χ ,

For s u p p o s e t h a t

F

and (A.4)

Μ

F'.

the F', the

does not meet

and by i t e r a t i n g

the

Appendix: filers Proposition u n i o n of subsets

(A.5).

subsets is

Let

is

principal.

Since

F^

on

Χ

MsF for

Such u l t r a f l i t e r s

a finite

filter

of

Fq.

Fg

an i n f i n i t e

formed

In f a c t

is

a refinement

or

X - M e F .

some

principal

filter

-If

the

t h e n one of

the

of

F^.

by

set

is

is

X

neither

principal.

necessarily a special

by t h e c o f i n i t e

For i f

X - Μ

(A.5).

generated

X

role

s u b s e t s of

is

any

is

Μ e Fg finite

ultrafilter

then

by

x,

F

either

and s o

This implies that

is

x.

any n o n - p r i n c i p a l u l t r a f i l t e r

But

χ e X - M,

X.

are c a l l e d

set every u l t r a f i l t e r

i s obviously not p r i n c i p a l

refinement

F

g e n e r a t e d by a g i v e n e l e m e n t o f

I n t h e c a s e of

p l a y e d by t h e

a member o f

on

F.

a l w a y s an u l t r a f i l t e r . In t h e c a s e of

be an u l t r a f i l t e r

Mj^, . . .

a member o f

The f i l t e r

F

(x) F

and s o we h a v e a

e F

is

the

contra-

diction . Proposition and is

Y

(A.6).

are s e t s .

an u l t r a f i l t e r For l e t

F

then

of

F

and

then

all

Ν

then

X.

an u l t r a f i l t e r ,

φ^Ρ

is

φφ~^Ν c Ν.

If

φ ^N

i s n o t a menber

Y.

space. forms

Y - Ν G

is

is

is

a

a member o f

a refinement

and s o

therefore

as

The c o l l e c t i o n o f

or

So i f G

a member o f

φ"^(Υ - Ν) = Χ -

Ν

of

φ^Ρ, φ^Ρ

hence c a n n o t be s t r i c t

and s o

asserted.

these preliminaries

in the sense

then

φ'^^Ν

Y - Ν /

χ

X

X

If

The r e f i n e m e n t

of

on

where

Y.

since

of

is a topological

be a f u n c t i o n ,

an u l t r a f i l t e r

Thus, e i t h e r

neighbourhoods of

is

Y

Y.

since

subsets

After X

on

F

Y - Ν £

Ν £ G

is

If

φ : X

be a s u b s e t o f

F.

Ν £ φ*F. φ*F

Ν

Ν « φ^Ρ

member of for

Let

we t u r n t o t h e

For each

situation

χ ε X

a filter

,

the family

Ν

of

the neighbourhood

neighbourhood f i l t e r s

t h a t each neighbourhood

where

of

χ

is

coherent,

contains a

filte:

I.M. James neighbourhood point

of

N'

N'.

on a s e t

X

such that

In f a c t

the

and

filter

is

a neighbourhood

a collection

determines

contains x,

Ν

a topology

(ii)

of

on

filters X if

t h e c o n d i t i o n of

i s then the neighbourhood

of

{

(i)

every

:

χ

e x)

e a c h member

coherence filter

of

of

is

satisfied;

χ

in

the

topology. Definition X.

(A.7).

A point

of

Let

X

is

F

be a f i l t e r

an a d h e r e n c e

point

a d h e r e n c e p o i n t o f e v e r y member o f It fied

is

sufficient,

of

b y t h e members of

point

of

F

then

F

for

a base for

F.

Obviously,

the adherence

points,

just

is

For e x a m p l e , cofinite

topology,

subsets.

F

if

it

is

an

the condition If

χ

is

t o be

an

satis-

adherence

a d m i t a common r e f i n e m e n t . set

of

F,

the i n t e r s e c t i o n

suppose

of

space

F.

course,

and

on t h e t o p o l o g i c a l

that

X

of

is

i.e.

an i n f i n i t e

set

adherence

t h e c l o s e d members o f

in which the c l o s e d

Then t h e a d h e r e n c e

the s e t of

of

sets

set with are the

every f i l t e r

F.

the

finite F

on

X

is

non-empty. Definition X.

(A.8).

A p o i n t of

ment o f

its

F

X

is

F

condition

converges

to

be a f i l t e r

a limit

neighbourhood

When t h i s that

Let

it

is

are i n h e r i t e d

is

if

x.

satisfied

by a p o i n t

Note t h a t

if

of

F.

by r e f i n e m e n t s ,

t h e o t h e r way r o u n d .

is necessarily

F

F

is

a

space

refine-

filter.

t h e n s o d o e s any r e f i n e m e n t points

p o i n t of

on t h e t o p o l o g i c a l

F

χ

converges

In other words, whereas for

To c l a r i f y

we to

the

say χ

limit

adherence

Of c o u r s e a l i m i t p o i n t

an a d h e r e n c e p o i n t .

e X

of

a

points filter

relationship

we p r o v e Proposition

(A.9).

soace

A p o i n t of

X.

Let

F X

be a f i l t e r is

on t h e

topological

an a d h e r e n c e p o i n t o f

F

if

and

Appendix: fillers only i f

it

is

For i f refinement

a limit X

is

of

F

and c o n v e r g e s

to

G

point

of

seme r e f i n e m e n t

an a d h e r e n c e p o i n t o f and t h e n e i g h b o u r h o o d x.

sane r e f i n e m e n t member o f

12"/

G

Conversely, of

F

if

F

χ

is

of

F. '

then the

filter

H^

a limit

F;

common

is

then each neighbourhood

and s o m e e t s e v e r y member o f

adherence point

of

defined

point

of

χ

is a

of

thus

χ

refinement

is

i s an

F.

In t h e c a s e of

an u l t r a f i l t e r

strict

ruled

o u t and s o we d e d u c e Corollary

(A.10).

space

each adherence p o i n t

X

The f o l l o w i n g

For an u l t r a f i l t e r

criterion

of

for

F

F is

on t h e

topological

a l s o a l i m i t p o i n t of

the Hausdorff

condition

is

F

often

useful.

Proposition space

if

(A.11).

and o n l y i f

each convergent {x}

is

the adherence

a Hausdorff

Let

filter

exist

ς,η

η

if

χ

s e t of

X

condition is is

is

a Hausdorff

satisfied

a limit point

of

by

F

then

F.

suppose

be d i s t i n c t ξ

η

converges

that

limit points

there exists

that

are unique

the s t a t e d c o n d i t i o n

p o i n t s of to

ζ

t,

and

suppose that a filter

as a l i m i t point.

every neighbourhood

π

for

with

X.

and s o ,

a s an a d h e r e n c e p o i n t .

n e i g h b o u r h o o d s of Conversely,

ζ,η

F:

in particular,

(A.11),

of admit

following

space

in

space.

To p r o v e

cannot

the

filter

This shows,

fied.

The t o p o l o g i c a l

The by t h e

η

and s o

satis-

neighbourhood condition,

In o t h e r w o r d s ,

there

which do n o t m e e t . seme p a i r of d i s t i n c t

ζ

points

a s an a d h e r e n c e p o i n t £md

Then e v e r y n e i g h b o u r h o o d o f of

is

X

ξ

meets

i s not a Hausdorff

C o n t i n u i t y c a n be n e a t l y c h a r a c t e r i z e d

i n t e r m s of

space. filters.

I.. James as

follows.

Proposition and

Y

(A.12).

are t o p o l o g i c a l

c o n d i t i o n for φ

on

Y

φ : X + Y

spaces.

F

converges

Sufficiency

on

X

to

φ (x) .

i s obvious,

suppose that

converges to .

X

then

But

so φ,Ρ φ(x),

χ

as It

F

continuity at

refines

^φ(χ)'

the

N^,

Ν . .,

at

that filter

F =

and

x.

To p r o v e

χ.

If

hence

o t h e r words

is

image

φ^^F

φ^F

F

refines

by c o n t i n u i t y

Φ ^X ^

X

sufficient

χ e X

χ

i s continuous

where

and

s i n c e we c a n t a k e

refines

χ

refines

F

at a p o i n t

converges to

obtain the usual condition for necessity,

be a f u n c t i o n ,

The n e c e s s a r y

t o be c o n t i n u o u s

whenever a f i l t e r j>,F

Let

at

x,

and

converges

to

required.

is

useful.

in relation to conpactness In f a c t

Definition

conpact

(A.13).

every f i l t e r

on

To r e l a t e

that f i l t e r s

s p a c e s can be d e f i n e d

The t o p o l o g i c a l

X

space

a d m i t s an a d h e r e n c e

this definition

X

are most

as

follows.

i s compact

if

point.

t o t h e more f a m i l i a r

one we

prove Proposition

(A.14).

The t o p o l o g i c a l

and o n l y i f

e v e r y open c o v e r i n g

For s u p p o s e t h a t Γ

of

X

ments of since of

we c o n s i d e r

is

diction,

Γ.

a covering.

of

Γ

covers

X.

that every f i n i t e

section.

Then

base after

taking

Γ*

satisfies

finite

compact

h a s empty

if

subcovering. covering

formed by t h e

comple-

intersection

t h a t some f i n i t e

subfamily

and h e n c e t h e

corresponding

For s u p p o s e , subfamily

is

admits a f i n i t e

Γ*

Then Γ* I assert

X

G i v e n an o p e n

the dual family

a l s o h a s empty i n t e r s e c t i o n ,

subfamily

X

i s compact.

t h e members o f

Γ

Γ*

X

of

space

of

Γ*

to obtain a

has non-empty

the c o n d i t i o n for

intersections,

contra-

a subbase,

and s o g e n e r a t e s a

interi.e. filter

a

Appenix: filers F

on

say

X .

.

By c o m p a c t n e s s ,

Now

particular

χ

belongs

F

a d m i t s an a d h e r e n c e

t o each of

t o t h e members o f

t h e c l o s e d members o f

Γ*.

Thus we h a v e o u r

C o n v e r s e l y suppose t h a t e v e r y open c o v e r i n g a finite

subcovering.

Suppose,

there e x i s t s a f i l t e r the is

intersection empty.

of

F

on

X

the family

Hence t h e d u a l

Γ*

of c l o s u r e s Γ

corresponding

subfamily

Γ*

Proposition F

Let

A

h a s empty i n t e r s e c t i o n . the f i l t e r

F

set

a contradiction,

that

e v e r y member of

the

on

V

of

F

Since

X

where

χ £ X.

X - V X - V

i s compact Now

G

and s o

and from

χ

admits

χ < A

χ e A,

(A.16) .

and o n l y i f

and s o we

of

(A.14).

of

a filter of

Suppose,

A

a filter

G

is

on

G,

of

i.e.

F,

Then X.

χ

the neighbourhood

t h e b a s e of

obtain

F.

an a d h e r e n c e p o i n t

since

to

say, V

of

A

the trace

since

G

w h i c h g i v e s u s our c o n t r a d i c t i o n .

of

refines Hence

obtain

The t o p o l o g i c a l

every u l t r a f i l t e r

When t h e a d h e r e n c e s e t we

A.

i s an a d h e r e n c e p o i n t

( A . 1 0 ) a b o v e we

Corollary

meets

generates

d o e s n o t m e e t t h e members o f

F,

But

Then e v e r y n e i g h b o u r h o o d

of

However

an

Then t h e

be a n e i g h b o u r h o o d

F.

F

F.

For l e t

trace

admits

o f members of

subcovering.

be t h e a d h e r e n c e

X.

X

point.

This completes the proof

on t h e c o m p a c t s p a c e

a manber of

of

of c o m p l e m e n t s forms

a r e a l l members o f

(A.15).

contradiction

Then

a finite

have our c o n t r a d i c t i o n .

in

w i t h no a d h e r e n c e

and s o a d m i t s

Γ*

F ,

that

open c o v e r i n g

t h e members o f

χ ,

to obtain a c o n t r a d i c t i o n ,

family

of

point

in

on

space

X

is

(A.15)

X

i s compact

if

convergent.

consists

of

a single

point

obtain

Corollary Then point.

F

(A.17)

Let

i s convergent

F if

be a f i l t e r F

on t h e c o m p a c t s p a c e

admits p r e c i s e l y

one

adherence

X.

Exercises

1.

Show t h a t

set

Ζ

Ε X X

for

given

of

integers

(η =

1,2,...)

prime

ρ

a uniform

s t r u c t u r e on t h e

i s g e n e r a t e d by t h e s u b s e t s

,

where

t

(ζ,η)

D^

D^

of

i f and o n l y

if

ζ Ξ η mod p*^ . 2.

Let

φ

tinuous for

be a r e a l - v a l u e d

on t h e u n i f o r m

some

e > Ο

continuous, 3.

Let

φ,

continuous

space

and a l l

where ψ

ψ(χ)

function X.

Suppose t h a t

χ e X.

Show t h a t

= (φ(χ))

be r e a l - v a l u e d

on t h e u n i f o r m

which i s u n i f o r m l y

functions

space

X .

4.

t h e uniform space

bounded i f η

for

such t h a t

of

each entourage Η c D"[S]

Show t h a t

the union of

5.

X

Let

and

t h a t the uniform space

Y^

not

discrete.

6.

Let and

D

of

X

spaces,

t o be

there e x i s t s

with

X

an

integer

S

of

Η .

is

bounded.

discrete.

Show

function-

pointwise

an example where t h e uniform

structure

spaces.

is

a uniform

7.

the uniform

structure

ψ

continuous.

said

subset

and

of

Γ:Χ-*·ΧχΥ Show t h a t

is

φ

of uniform c o n v e r g e n c e on t h e

be a u n i f o r m l y

are uniform

X

uniformly

if

i s uniformly

some f i n i t e

be u n i f o r m

and g i v e

uniformly

which are

a p a i r of b o u n d e d s u b s e t s

structure

φ : X ->· Y Y

φ.ψ

c o i n c i d e s w i t h the uniform

convergence,

X

Y

for

is

Sliow t h a t

product

Η

ψ

a e

^ .

are bounded t h e n t h e i r The s u b s e t

|φ(χ)|

con-

continuous

function,

Show t h a t t h e g r a p h

is

where

function

embedding. space

X

is

totally

bounded i f

and

Exercises only

if

all

8.

In t h e uniform

connected uniform

countable

subsets

C^

t o be r e l a t e d

if

η

(ζ,η)

tains

Let

that

for

for

all

11·

D

space

R

Show t h a t

12·

Let

space

X/R

R

Let

uniform relation

X

partitions.

deduce

that

Β

Let

of

that

for

the Euclidean

η

exists

by

C^ .

are

said

an

equivalence

integer classes

connected.

a given point

the u n i o n of is

both

χ

con-

is

the

closed

equivalence

X ,

.

for

every

structure

φ(t)

χ

uniform

,

in

X .

Sj^

(πχπ)

and g i v e

topology.

uniform

a

on

compatible

^S .

defined is

structure .

t X .

relation is

structure

= t3

uniform

on t h e

equivalence

ultrafilter

the uniform

d"IiiJ

on t h e

relation

S„ = κ

the uniform

Show

sets

all

Show t h a t

where

X .

relation

in the quotient

X/R

X

space

o p e n and c l o s e d

equivalence

be g i v e n by

uniform

and

there

X

R[xJ

space

infinite

ξ

the uniform

Show t h a t

φ : ]R -