Local compactness in approach spaces I

VOL. 21 NO. 3 (1998) 429-438

LOCAL

COMPACTNESS IN APPROACH SPACES

R.LOWEN

DepartmentofMathematics UniversityofAntwerp, RUCA

Groenenborgerlaan 171, B-2020Antwerp, Belgium

C.VERBEECK

Department

ofMathematicsandComputerScience University of

Antwerp, UIA

Universiteitsplein 1, B-2610 Wilrijk, Belgium

(Received August 14, 1996)

ABSTRACT.This paper introduces some definitionsofa concept of local compactnessin ap-proach spaces. The basicrelationships between theseconceptsis studied, and measuresof local compactnessaredefined.

KEY

WORDS AND PHRASES: Approach space, measure of compactness,

(basis-)

local compactness, measure of

(basis-)

local compactness.

1991AMS SUBJECT CLASSIFICATION CODES: 54A05;54D45.

1. INTRODUCTION

2. PRELIMINARIES

Given a set

X

wedenoteitspowerset by 2x and thesetofits finitesubsets by2(x). We recall those concepts andresults from

Lowen

[2,

3]

which we require inthesequel.

A

map

:Xx 2x

[0, c]

iscalled adistance if it fulfils

(D1)

VA

2X,Vx

X

x

A

5(x,A)

O.

(D2)

Vx X:

5(x,0)

(D3) VA,

B

2

x,

Vx

X:

5(x,

A

t2

B)

5(x,

A)

A

6(x,

S).

(D4)

VA

2X,Vx

X,

Ve

[0,

o]

:6(x,A) <_

5(x,A()) +

e where

A

() _:=

{x[5(x,A) <_ e}.

If is adistance on

X

and

A

C

X,

thefunction

6A

X

-

[0,

oo]

isdefinedby

5A(x)

:=

5(x,A).

A

collection

(,4(x))xex

ofideals in

[0,

o]

x is calledanapproachsystem ifit fulfils

(A1)

Vx

X,

V

4(x)

v(x)

0.

(A)

W

e X,

V

e

[0,

(V,

N

]0, oc[,

3

,4(x)

v

AN

_<

+

e)

=

o

A(x).

(AS)

Vx

X,

Vo

A(x),

VN

10,

3(v,),ex

I]

A(z),

Vz,

y X:

o(y)

AN

< x(z)

+

o,(y).

zEx

The elements ofan approach system are called localdistances. For ease in notation we shall,

wheneverconvenientdenote anapproach system

(A(x)),ex

also simply

A.

If

A

isanapproach systemthenA:=

(A(x))ex

iscalleda basisfor

A

if itfulfilstheproperties

(bl)

Vx

X

A(x)

is abasisforanideal.

(b2)

Vx X:

.A(x)

=/it(z)

where

/(x)

:=

{

[0, oc]

x

V,N

]0,o0[,=t

h(x):

^N_<

+e}.

PROPOSITION2.1

[3]

1.

If

4is anapproach systemon

X

thenthe map

6t"

X

x 2x

--

[0,

oo]"

(x,

A)

sup inf

qo(a)

isadistance onX.

$.

If

6is a distanceonX thenthesystem

At

where

for

all x

X

A,(x)

{

[0,

o01

x VA

c

X-inf

o(a) <

5(x,A)}

aA

3..A.

.,4 and

6.a

6.

A

set

X

equipped with anapproachsystemorequivalentlyadistance iscalledan approachspace and is usuallydenoted

(X,

J

l).

Theassociated distance is usually denoted simply 6 instead of

6A

andanalogously if5isthe primary defined structure,

A

isusuallysimplydenoted,4,unless confusion mightoccur.

PROPOSITION

2.2

[3] If

A

is abasis

for

,4 then

.

isalso obtainedby

(,a)

sup

If

(X,

,4)

and

(X’, 4’)

areapproachspaces and

f

X

X’

is amap then

f

iscalledacontraction ifitfulfilsanyofthefollowing equivalent

(see

Lowen

[3])

conditions:

(el)

Vz

X,

Vo’

A’(f(x))

o’

o

f

(c2)

Forany basish’for

A

r,

Vx

X,

re’

A’(f(x))

’

o

f

e

A(x).

(c3)

Vx

X,

VA

C

X:

’(f(x), f(A)) <_ (x,A).

Approach spaces and contractionsformatopological construct Lowen

[3]

which wedenote AP. Forcategorical concepts, inparticular topologicalcategories, werefertoAdmek et al.

[4].

TOP canbe embeddedasabireflective and bicoreflective subconstruct of

A

P.Theembeddingsfunctor isgiven by

(X,

7")

--+

(X,

AT)

leavingmorphismsunaltered and where

AT

istheapproachsystem

AT(x)

:=

{o

[0,

c]

x

o(x)

0,

o

u.s.c,at

x}

foreveryx

X,

andwhich hasas basis the collection

{Or

Vneighborhood ofxfor

T}

for every x

X,

where

Or(x)

0 if x V and

Oy(x)

cxif x V. The associateddistanceisgiven by

6(x,

A)

0ifx and

6(x,

A)

if x

.

The subconstruct thusobtained isisomorphicto

TOP.

Werecall thatthebicoreflection of

(X,A)

in TOP isgivenby

idx

(X,,4-a)

-+

(X,

J

I)

where

7

isthetopologyon

X

determined by the neighborhood system

orequivalently bythe closureoperator :=

{x

X

(x,

A)

0}

forevery

A

2

x.

The construct

pq-MET

of extended pq-metric spaces and non-expansive maps too can be embedded as abicoreflectivesubconstruct ofAP. The embeddingis given by

(X, d)

--

(X,)

leaving morphisms unaltered and where

Ad

istheapproachsystem

J[d(x)

:=

{o

[0,

]x

[o _<

d(x,-)},

Vx

e

X

with obvious basis consisting of the single element

d(x,

.).

As to be expected the associated distance isgiven by

6(x,

A)

infaea

d(x,

a).

As

for topological spaces aconvergence theorycanbe developedin

AP

(see

E.

and

R. Lowen

[5,

6]

formore

details).

The difference withtopological spaceshowever isthatwitheach filter and eachpoint we can give a distancethepoint "is awayfrom beingalimit point"of thefilter. Precisely thisgoesasfollows. Givenaset

X,

F(X)

istheset ofall filters on

X;

if

if

G

consistsofasinglesetGwewrite

stackxG

andifmoreoverGconsistsofasingle point a,we write

stackxa

forshort. Ifno confusion canoccur,wedrop the subscript X. Also,if

F

C

X

we abbreviate

U(stack F)

by

U(F).

Theset sec

"

isdefinedas the unionof all ultrafiltersfinerthan

’,

which means sec

"

:=

{A

C

X

VFE

"

A

A

F

#

0}.

Let

(X, 5)

be an approach space and

:F

F(X),

then thelimit

(-function)

of isdefinedas

’(z)

:= sup

(f(z, A),

Vz X. AEsec

Itwillbe usefulalsotohave a descriptionof

6

not in termsof6 butin termsof the associated

approachsystem

t.

PROPOSITION2.3

[7]

Let

(X, .A)

be anapproachspace. Forany

yz

E

F(X)

andx EX wehave

AAgV(x)=

sup inf_supo(y)

whereboth.4 and

.T

maybereplaced by bases.

Itis worthwile to mentionthatlimits also asin TOPprovide yeta third way to describeapproach spaces

(see

E.andR.

Lowen

[5]

formore

details).

Forusitsufficesto mention thatthe distance 5can berecovered from

6

by

(x,A)=

inf

)lJ(x).

uu(A)

Convergence in the topological bicoreflection

(X, 7)

ofan approach space

(X,

5)

can easily be derived from the limitassociatedwith5. If

"

F(X)

then

"

-

xin

T

ifand onlyif

(x)

0.

In

the case of cx)-pq-metric spaces the associatedlimittakes on a more simple and intuitive form. If

"

E

F(X)

then

Y(x)

infsup

d(x,

y),

F. yF

andincase

"

isgeneratedbyasequence

(x,),

thissimplymeansthat

’(x)

limsup

d(x,

DEFINITION 2.4 Given anapproachspace

(X,A),

we

define

themeasure

o.f

compactness

oI

X as

#c(X)

sup inf

/(x).

uev(x)ex

The idea behind this definition is the following. Compactness means every ulrafilter should haveaconvergencepoint. Thereforetheinformationgivenby#eisbasedonthe verification for all ultrafilterswhat are their "bes

convergence"

points. Beforeallelsewegiveanumberof equivalent forms ofthisdefinition.

PROPOSITION2.5

[2]

Foranyapproachspace

(X, riO,

we have

(X)

sup inf sup inf

(x)(z)

pEI’I=x

A(=)YE2(x) zEX

sup inf supinf

(x)(z)

cEl-[=x

A(x) YE2(X) zEX xEY

THEOREM2.6

[2]

1.

If

(X, AT)

is atopological approach space then

(X,

T)

is compact

if

andonly

if u(X)

O. 2.

If

(X,‘4d)

is ano-pq-metric approach space then

(X,d)

is totally bounded

if

and only

if

u(x)

=o.

3.

If

(X, ‘4d)

is apq-metricapproachspace then

(X,

d)

isbounded

if

and only

if #(X)

<

.

THEOREM2.7

[2]

1.

If

(X,A)

and

(X’,A’)

are approach spaces and

f

(X,A)

(X’,A’)

is a surjective contractionthen

#e(X’) < #e(X).

2.

If

(X,

Aj)Ij

isafamily

of

approach spaces then

3. SOME NOTIONS OF LOCAL COMPACTNESS IN

AP

In this sectionwe willdefine somenotions of localcompactness and basis-local compactness in

A P,

which ontopological spacescoincide withthe topologicalnotions of local compactness and basis-local compactness. Wewilldenote thenotions of local compactness by LCn wheren isa number between 1and5,and theassociated notionsof basis-local compactness likewise byBLCn.

DEFINITION3.1 Let

(X,,4)

beanapproachspace.

1.

(X, ,4)

isLC1

if

andonly

if

itstopological

coreflection

islocallycompact. 2.

Define

(x,.)

Lee

==

Yx

X,

VY x,3F

Y #(F)

0 ,=,

v

e

x, v

e v(): o(v)

o,

where

])(x)

isthe neighborhood

filter

of

x in thetopological

coreflection.

Define

(X,

.,4)

isLC3

Yx

X,.T

x- inf

(F)

0

F{"

==

VxX"

inf

#c(V)=O

vv(x)

,=, Vx

X,

Ve

>

0,3iv

A(z),3

>

0:

({

<

6}) <

.

4.

(X,‘4)

is

LC4

if

and only

if

VxX,’v’e>O:

inf

(F)<e,

FeV,(x)

5.

(X,

,4)

isLC5

if

andonly

if

VxX,

Ve>0" inf

pc(F)<

inf

It iseasily verifiedthat thegiven definitionsofLC2areequivalent, andlikewise arethoseofLC3.

Wehavethefollowingobvious relations betweenthedifferentLCn.

PROPOSITION3.2 LC1 LC2 LC3

LC4

LC5.

PROPOSITION3.3 Let

(X,

T)

be atopologicalspace. Then

(X,

T)

is locally compact

(X, AT-)

is

LC

(X, AT-)

isLC5.

This is astraightforward result and it shows the LCn can be considered as generalizations of topological local compactnessin the context ofapproach spaces. Noticethat for atopological space,

Ve(x)

V(x)

foreverye

>

0andevery x X.

PROPOSITION3.4 1. Let

(X,

d)

beanoc-pq-metric space. Then

(X,

d)

isLC2

if

and only

if

everypointpossessesatotally bounded neighborhood.

2.

Every

c-pq-metricspaceisLC3and

LC4.

PROOF.

1. Let

(X, d)

be an x-pq-metric space. Then

(X, d)

isLC2ifand only ifVx

X,

3V

l)(x)

#c(V)

0. Denotetheopen ballwith center yand radius eby

B(y,),

then

pc(V)

0if and onlyifVe >0, 3Y 2(v) VC

[.Jy

B(y,

),

ifand onlyifVistotallybounded. 2. SinceLC4impliesLC3weonlyhave to showevery c-pq-metricspace

(X,

d)

isLC4. Now

(X,

d)

is LC4 ifand onlyiffor everyx

X

and every e

>

0,

inffev,(x)pc(F)

_<

.

But

B(x,e)

belongsto

])e(x)

andby adapting the first part of theproofwesee

#c(B(x,e)) <_

.

Forevery

LCn,

wewill introduce acorrespondingnotionBLCnof basis-local compactness.

DEFINITION3.5 Let

(X,.4)

beanapproach space.

1.

(X,A)

isBLC1

if

andonly

if

its

TOP-coreflection

isbasis-locally compact. 2.

(X, ,4)

isBLC2

if

and only

if

Vx

X,

VV

12(x),

SW

V(x)

WCV and

#c(W)

O. 3.

(X, ‘4)

isBLC3

if

and only

if

VxX-

sup inf

#(W)=0.

VV(x)wv()

WCV

4- (X,A)

is

BLC

if

and only

if

VxX,

Ve>0,VFre(z),Ve’<e-

inf

#c(G) <

GE(z)

5.

(X,

.,4)

isBLG5

if

and only

if

Vxfi

X,V >

O, VF

V(z)

V’

<

e" inf

#(G)

<

inf

SVe(x)(y).

GCF

Ifan approh spaceis

BLCn,

then it is alsoLCn.

In

order to see this, notice that we have

Ye(x)

,<e,(x)

for

eve

x

X

and

>

0. The relationsbetween thedifferentBLCnare for theLCn:

PROPOSITION 3.6 BLC1 BLC2 BLC3

PROPOSITION3.7

Let (X,

T)

beatopologil

sce.

Then

(X,

T)

bas-locally compact

(X, AT)

BLC1

(X,

Av)

BLC2

(X,

AT)

BLC3

(X,A)

u

BLC4

(X,A)

isBLC5.

Thisrultillustrates theBLCncan beewed generalizations of topological bis-local com-ptnessinthe context ofapproachspaces.

For --metric

spaces, weget the following result.

PROPOSITION3.8 1. Let

(X, d)

be an -pq-metric space. Then

(X, d)

andonly i/everypointpossessesaneighborhood bcoisting

of

totally bounded neighbor-hoo&.

2.

Eve --metc

spaceisBLC3 and

BLC.

For

some notionsof local compactness in

AP

we can introduceameureof local comptns whichisageneralizationinwhichall thenice properti arepreserved.

DEFINITION3.9 Let

(X,

A)

beanapproach space.

Lc3(X)

:= sup inf pc(V). xvv()

UBZC(X)

:=sup sup inf

xX V()wv()

WCV

:=

(

v

o.

zeX>0

.BLc4(X)

:=supsup sup sup

(

inf

,c(G)

e’)

VO.

GCF

U5cs(X)

:=supsup

(

inf

u(F)

inf

AV(x)(y))

V0. xx>0

Lcs(X)

:=supsup sup sup

(

inf

c(G)

inf

AV,,

(x)

(y))

v

0.

xx>0FV,(x)’< av(,) yx

GCF

Forn

{3,

4,

5},

()Lcn(X)

is called the meure

of (b-)

local compactness ofX. This is justified by thefollongproposition.

PROPOSITION3.10

I

(X,A)

anapproach spaceand n

{3,

4,

5},

then

()Lc(X)

0

if

and

oy

if (X,A)

(B)L.

The following

inualities

caneilybecheck.

PROPOSITION 3.11

Let

(X,A)

be anappwach space. Then 1. Forall n

4. COUNTEREXAMPLES

Proposition3.2 gives some relationsbetween the differentnotionsof local compactnessin

A

P.

In

thissection,we willshowthese relations arereally allthere is tofind,exceptperhaps LC4

=

LCb. 1. Notethat thereexist approach spaces which are not LC3, for instance, every topological

spacewhichdoes notpossess localcompactness.

2. To prove LC2

:)

LC1,just considerthemetricspace

(d,

d)

ofrational numbers with the euclidean metric. Everypointpossessesatotally bounded neighborhood, so

((,

d)

isLC2.

Butclearly the topological coreflectionisnotlocally compact,so

((, d)

isnot LC1.

3. We shallnow show LC5 LC2. Tothis end, consider the metric

(Hilbert)

space

(12,

d).

Since it is a metricspace,

(l2, d)

is LC4. Wealready know that for allx6 and alle

>

0,

infet2

Ar(x)(y)

<

6. Weshall prove that also

inft2

Ar(x)(y)

>

6:

infAVe(z)(y) inf sup

6(y,A)

Y6-.12 Y12 A.secVe(x)

inf sup inf

d(y,a).

Y6--12ACB(z,e)#$ a6..A

Nowtake y6

12.

Ify

9

B(x,e),

defineA:=

{x},

thenitisobviousthat

ACIB(x,e)

:/:.

and

infaeA

d(y,a)

d(y,x)

>_

6. Ify 6_.

B(x,e)

\

{x},

we canfind some

y’

6

B(x,e)

such that d(y, y’)

_>

e. In thatcase,choose

A

:=

{y’}.

Ify x then _supanB(,e)#

infae

A

d(x,

a)

6.

Hence

wecanconclude

infet

AVe(x)(y)

e. This implies

(/2,

d)

isalso LCb. Sinceno point in

(/,

d)

hasatotally boundedneighborhood,

(/,

d)

isnotLC2.

4.

In

order toseeLC1

LC4,

considerthe following approach space

(X, A):

Let

X

beaset,

-

afilter onXand

f"

X

[0,

oo]

afunction. If

;

isafilter on

X

and x6

X,

we define

f(x)

if

Y

stackxC

O

and

;

#

stackx

Ag;(x)

A(j);(x):=

oo if

’

stackx

g

;

0 if

;

stackx.

In

E.andR. Lowen

[6]

itwasshown that

(X,

A)

is an approach space.

Moreover,

if

H

isan ultrafilteron

X

andx6

X,

f(x) if’CHandH#stackx

AH(x)

o if

"

/2

and

H

#

stack x 0 if

H

stackx.

Ifthere exists an x6XsuchthatH stackx,theninf,x A/g(x) 0. If for every element x

e

X,

H

#

stackxand

Y

C

,

then inf,ex

AH(x)

infzex f(x).

Iffor every elementxof

X,

/4

#

stackxand

"

fZ

H,

then

infzx

A/4(x)

oo.

So

weget thefollowingthreecases. If

X

is finite

(i.e.,

all ultrafilters on

X

are point

filters),

then

#c(X)

supuev(x) inf,ex

AH(X)

0. If

X

is infinite and iffor every ultratilter

H

on

X,

not being a pointfilter,

"

C

H,

then

p(X)

influx f(x).

Finally, if

X

isinfinite and there exists some ultrafilter/4 on

X,

not a point filter, such that

"

if

H,

then

#(X)

oo.

Let B

be the intersection of all ultrafilters on

X

which are not a pointfilter.

A

straightforward verification shows that

B

{X

\

A A

is

finite}.

Usingthis

fact,

wecanstate 0 if

X

isfinite

Inthesequelwe willsupposeXisinfinite, is anultrafilter on

X

withonlyinfiniteelements, and

f

o. Then for every ultrafilterL/on

X,

f(x)

if’=L/

AL/(x)

if

.T /2

and

L/#

stackx 0 if/,/=stackx.

Inorder to know

VE(x)

we need to find

A(x).

A(x)

*

Vbl

U(X)

supinf

(z) < M2(x)

U EblzEU

supin[

(z) < f(x)

and sup inf

(z)

0

FzF _Ustk zU

supinf

(z) </(x)

and

(x)

0.

FzF

Weinfer

V(x)

stack

{{qa

<

e}

sup inf

(z)

<

f(x)

and

q(x)

0}.

F6.. z6.F

Notice thatfor anyGC

X

the initialstructureonGis givenby

A(yG,I,a

andconsequently the expressionfor

#c(G)

isobtainedby replacingXbyGinthe expressionfor

#c(X).

Hence weget

inf

#c(G)

0 ifthereexists somefiniteG

V(x),

inf

#e(G)

inf inf

f(z)

if everyG

VE(x)

{C\A Aisfinite}D.’l

isinfiniteand thereexists someG

l)(x)

such that

(G \

A

IA

is

finite}

D

’IG,

and

inf

#c(G)

oc if every G

Ve(x)

isinfiniteand satisfies

aev(x)

{G

\

A[A

is

finite}

;fi

-O-Considersomex X suchthat

f(x)

<

.

Ifwetakee

>

f(z)

then

V(z)

containsonly infinite elements: ifqa satisfies

supF.infzFp(z

< f(x)

and

(x)

0, then for every

F

"

and for everyO

>

0,there exists some z

F

such that

(z) < f(z)

+ g,

so

{

<

}

isinfinite, for

"

is an ultrafiltercontainingonly infiniteelements. The factthat

"

is an ultrafilter also implies that

{G

\

A

A

is

finite}

-G

for every G

V(x),

So we can conclude

infGv,(x)#,(G)

c

>

s, whence

(X,

A)

isnotLC4.

Now choose a function

f

as before, but moreover satisfying

f

>

0.

For

every x

X,

(z)

stackx, so

{x}

isacompactneighborhood ofz. This means

(X,

)

isLC1.

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