COMPACTIFYING A CONVERGENCE SPACE WITH FUNCTIONS

COMPACTIFYING A CONVERGENCE SPACE WITH FUNCTIONS

ROBERT P.

ANDRI

Luther

College

UniversityofRegina

Campus

Regina, Saskatchewan Canada

$4S0A2

e-maih

andrer@

meena.uregina.ca

(Received November 22, 1994 and in revised form March 25,1995)

ABSTRACT. A convergence

spaceis asettogetherwitha

convergence

structure.

In

this

paper

we discuss amethodofconstructing compactificationsonaclass of

convergence spaces by

useof functions.

KEYWORDS AND PHRASES:

Compactification,

convergence space, pretopological,

singular compactification, singularsetofafunction.

A.M.S. SUBJECT CLASSIFICATION CODE:

54D35, 54A20, 54B20.

1.

INTRODUCTION.

Theterms whosedefinitions are given here for the sake of

completeness

are discussed in

many

textbooksin

topology. A

set

A

is a directedsetifthereexists a relation

<

on

A

such that

1) 5 < i

forall

5

e

A, 2) 51 <- 52

^and

52 <-

implies that

51 ^<

^and

³⁾

^if

il

^and

52

^belongto

^A

^{thenthereexistssome}

element

3

in

A

such that

51

^<

3

^and

2

^<

. ^A

^netinaset

^X

^isa function s

A -- ^X

^{froma directedset}

A

into

X.

If

.

^{is inthedomain}

^A

^{of thenets}

^{A X}

^{we willdenote}

^{s(,) by s.}

^andthenetsin

^X

^by

^s.

k

e

A }. For

a directedset

A

we willdenoteby

txA

^theset

ⁱ

^e

^{A i >} IX ^}.

^If

E

isasubset of thedirected set

A

then

E

is

cofinal

ⁱⁿ

^A

(orfrequentlyinA)if

IXA ^{E O}

^for

^any Ix ^A.

^If

^E -- ^X

^{isa function}

from into

X

then isasubnetof s

A - ^X

^iffor

^{any Ix A}

^thereexistsa

⁵

^e 1 such that

t[15:]

s[IXA]. ^A

^universalnet

(or ultranet)

isanetwith no

proper

subnet. The followingideasare introducedin

So [18]. A

convergencestructureon aset

X

is aclass

C

oforderedpairs (s,x) wheresisanetin

X

andx

X

such that for

any

(s,x) in

C

theordered pair (t,x) also belongs to

C

if is asubnet of s.

A convergence

space (X,C)isaset

X

on which we havedef’meda

convergence

structure

C.

Ifa

convergence

structure

C

isdefined on aset

X

wewill

usually

abbreviate

(X,C) by X.

Also the

phrase

s

converges

tox (denoted

by

s

--

^x)willmean

^(s,x)

^e

^{C. A} convergence space X

is compact ifeverynetin

X

has a convergentsubnetin

X

and, finally,

X

is

Hausdorffif

^nonetin

^X

convergestotwodistinctpointsin

X.

Throughoutthispaper

X

willdenotea

convergence space.

If

E X

then

clxE ^{E u {x}

^e

^X

^thereis

somenets in

E

such that s x

}. Note

that thisclosure operator isnotnecessarily idempotent,i.e.,

clxE

may

be a

proper

subset of

clxclxE. ^A

^subset

^E

^of

^X

^isdensein

^X

^if

clxE ^X.

^{If fisa}

^map

^from

^X

^intoa

convergence space

Y

then wesaythatf iscontinuousif s--4xin

X

impliesthat fos f(x).Furthermore, if f is one-to-one, continuous, and onto Y and if ^x- Y

-- ^X

is continuous then fis called a homeomorphismfrom

X

onto

Y. As

fortopological spacesa compactification

Y

of

X

is an orderedpair (Y,h)whereYis a compactconvergencespaceand h is ahomeomorphismof

X

into

Y

such thath[X]is dense in

Y.

Givenacompactification ctXof aspace

X

theoutgrowth (orremainder) of

X

in

X

is^txX’O(.

Two

compactifications

ctX

and

/X

^of

X

are saidtobe equivalent if there exists ahomeomorphismbetween 0tX and

"fX

that fixes thepointsof

X. We

willsaythat

X

ispseudotopological atxif

X

satisfies the followingproperty: ifeveryuniversalsubnet of anets in

X

convergestoxthen sconverges^tox.

We

will satthat

X

ispretopologicalatx if

X

satisfiesthefollowingproperty: If for anetofnets

S {ss 5

e

A}

eachnet

ss _{{ss I.t}

^{e Ai} (where}

As

^isthe domain ofss)convergestoapoint

xs

ⁱⁿ

^X

^and

^{x

^e

^A}

5

A,

I.t

^e

As}

orderedlexicographically by

A,

thenby

As,

convergestoapointxinX,then thenet

s

has asubnet which

converges

tox(i.e.

S

has a"diagonal

net"

that

converges

^tox).

A

convergence

space X

is saidtobepseudotopological (pretopological)if

X

ispseudotopological (respectively pretopological) atevery pointin

X. It

isknown that if aconvergence space

X

isbothpseudotopologicalandpretopological andsatisfiesthe property "foranet A

--

^X,

^s

^{xfor each}

⁵

^e

^A

^impliess

^s --

^{x",thenweobtain a}

topologyon

X by

definingthe closure of aset

E

in

X

as

clxE

^{x e}

^X

there is somenet in

E

such that

x}(see

1D of Willard

[20]).

Thefollowingtheorem isstraightforward.

THEOREM

1.

A

convergence space

X

is compact if andonlyifeveryuniversalnetin

X

converges.

We

willsay thatanets

{ss 5

^e

A}

in

X

iseventuallyin

E X

if

s[ktA]

K

E

for some

I.t

^e

^A.

^The followinglemma isProposition 3.3in

Aarnes

etal.

[2].

LEMMA

2. If s is anetin

X,

then s is universal if andonly^iffor each subset

E

of

X,

s iseventuallyⁱⁿ

E

or

eventually

inXkE.

In

So

[18]

the authordevelopsamethodforconstructingtheone-point compactificationof a non- compact Hausdorffconvergencespace

X

and discusses some of thepropertiesof thiscompactification.

In

thispaperwe discuss a

general

methodof constructing compactificationsof aconvergence

space

X.

In

particularwe usethismethodto constructacompactificationtowhich

every

real-valued boundedfunction on

X

extends.

2.

PRELIMINARY DEFINITIONS AND RESULTS.

Thefollowing techniqueforconstructing compactificationsismodeled on a methodof constructing Hausdorff compactificationsoflocallycompact Hausdorff spaces by usingfunctions from

X

intoa compact Hausdorffspace

K

(seeAndr6[1], Chandleretal. [5],[6],^Cainetal. [4],andFaulkner

[11]).

Let

f

X - ^K

be a continuous function from the non-compactHausdorff

convergence

space

X

intoa compact Hausdorff topological space

K. Let Y cl:f[X], Kx ^{{F X F}

^is

^compact}

^and

^S(f)

c{clvf[XkF] F Kx ^}.

The subset S(f)in

K

willbe called thesingularsetof f.Clearly

S(f)

^isclosed and hence is compact in

Y.

LEMMA

3.

Let

f

X K

bea functionfromanon-compact Hausdorff

convergence space X

into a compactHausdorff topological

space K.

Ifs

A -- ^X

^isanetin

^X

^thatdoesnotcontain a convergent subnet thenanysubnet of thenetfos in

Y clKf[X] converges

toapointinS(f).

PROOF. Let

f

X - ^K

be a function from a non-compactHausdorff

convergence

space

X

into a compact Hausdorfftopologicalspace

K

and let s

A - ^X

^{be anetin}

^X

^{that doesnot}contain a convergent subnet. Since

K

is compact thenetfos has a convergent subnet thatconvergestosomepointyⁱⁿY.

We

claimthaty S(f).

Let F

be a compact subset ofX.Since shas no convergent subnet in

X

there exists ala

A

suchthat

s[ktA]

XW. Consequently fos[gA] f[XXF]. Itfollows thaty^e

clvfos[I.tA]

clvf[XXF].

Since

F

was anarbitrarycompact subsetof

X,

y

n{clKf[XW] F Kx} S(f)

as claimed.

I’-1

3.

THE MAIN RESULTS.

Given anarbitrarycontinuousfunction f"

X

^--)

K

from a non-compactHausdorff

convergence

space

X

into a compact Hausdoffftopological space

K

let

X X u

S(f).

We

define a

convergence

structureon

X

asfollows

A

nets in

X converges

toapointxin

X

ifand onlyif s is

frequently

in

X

(i.e.,shasa cofinal subnet inX)and

six ^converges

^tox.

^Let

^f*

^X

^-o

^K

^{be the}function such that

f’Is(f)

^isthe identity functiononS(f)and

f*lx

^{f on}

^{x. A}

^{nets in}

^X f.converges

^toapointyin S(f)ifandonlyifshas no convergent subnet in

X

andf*os convergesto

y

inS(f) (noting that, bylemma3, y belongstoS(f)).

Letus nowverifywhether we have defined a

convergence

structureon

X f. We

arerequiredtoshow that if s

converges

tox in

X

and is asubnet ofsthen also

converges

tox.

It

will sufficetoshow this for anets in

X

that

converges

toapointx inS(f). Ifs isanetin

X

that

converges

toapointxin

S(f)

then shas no convergent subnet in

X

andf*os

conve:ges

toxin

S(f). Let

be a subnetofs.Thenf*ot is a subnet of f*os in

K

and so f*ot convergestox in

K;

hence converges tox.

It

follows that

X

is a convergence space.

Thefollowingisageneralizationof theorem 1.1 of Cain[4].

LEMMA

4. Letf be a continuous function from a Hausdorffconvergence space

X

toa compact Hausdorfftopological

space

Z.

Let Y clzf[X]

and

Kx ^{F

^c_

^{X F}

^is

^compact}.

^{Thentheset x}

^K

clxf[U]

isnotcompact foranyopen neighbourhood

U

ofxin

K} S(f) (= c{ clvf[XW] F Kx ^}).

PROOF. Let T {x K clxf-[U]

isnotcompactfor any open neighbourhood

U

ofxin

K }. We

will firstshow that

T

^c_

S(f). Let F Kx. ^{Suppose p belongs}

^to

^Ylvf[.

^{Thenthereexists an}

^open

neighbourhood

U

ofpin

Y

such that

f[U]

^g7

F

(since

Y

isa compactHausdorff topological space).

Hence

p

T

(since

clxf[U]

is compact).

We

have thus shown that

T

^g7

clvf[XkF].

Since

F

was arbitrarilychosen in

Kx0

^{itfollowsthat}

^T {clvf[X’xF] F

^e

Kx S(f). Suppose

now that x

belongs

toS(f). Ifx

belongs

to

YT

then thereexistsan

open

neighbourhood

U

ofxin

Y

suchthat

clxf*--[U]

is compact.

But

x

{clvf[XkF]" F

e

Kx}

clvf[Xlxf[U]]

(since

clxf-[U] Kx)

c::_

clvf[Xkf-[U]]

clvfof-[YkU]

YkU

Thiscontradicts that xbelongsto

U.

Consequently

{clf[X] F Kx} T.

The lemmafollows,

l-I DEFINITION

5.

We

will

say

thata

convergence space X

isa

LC space

if it satisfiesthe following property:

LC" Let S

s"

;

_{A be any}netofnetsin

X

suchthateachnet

ss ss

^e

^{A (where A}

^isthe

domain of

ss)

in

S

has no convergent subnet in

X. Let D Isn’t;

^e

^A,

^{e A}

^s]

^ordered

lexicographicallyby

A,

then

by .

^{Thennosubnet of}

^D

is compact.

PROPOSITION

6.

A Tychonoff

topologicalspace

X

islocallycompact ifand onlyif

X

isan

LC space.

PROOF. Suppose X

isa

locally

compact

Tychonoff space. We

canthenconstructthe

Stone-ech

compactification

[X

in which

X

is

open

(see 18.4ofWillard

[2@]).

Let

S

s;"

6

e A be anetofnets in

X

such that eachnet

s s

^e

^A]

^hasno convergentsubnetin

X. Let D s

Suppose

that,for eachie

A,

l(t)isthe limit ofsome convergent subnet ti

={si ^IX

^{Ei} ofsi. Since}

13X%X

^iscompact the^net {l(ti)

5 A}

hasa subnet{l(ti)

5 E}

^whichconvergestosomepointxin it"

5

e

Y., _tx Z5}

(itself a subnet ofD).Then

T

isof the form

T {si

it.

I3X. ^{Let T}

^{beanysubnet of}

it. it.

ie A,

IX

A} (where {si" ie A}isasubnet of{si"

5

e

Z5}

for each

8 X;). It

follows that

{s5 ^IX

^Ai}

^converges

^to l(ti), for each

8

e A. Since

I3X

^is

topologicalit ispretopological. Hencethenet

T s ⁵

^{e A,}

^IX

^e

^Ai

has a subnet

H

thatconverges^tox (since {l(ti)

A

convergestox).

It

thenfollows thateverysubnet of

H converges

tox,i.e., no subnet of

H converges

in

X.

Thismeansthat the subnet

T

of

D

has a subnet

H

with noconvergent subnet inX.

We

have shown that

X

is a

LC space.

Wenow

prove

the converse.

Suppose

Xis aTychonoff LC spacethatisnotlocallycompact. Then the

outgrowth X’XX

^ofthe

^Stone-ech

compactification

13X

^of

^X

^{isnotclosed in}

13X

^{(see 18.4of}

^[20]).

It"

5

e

A, IX

^{Ai},where si}

Thenthereexists anets in

I3XXX

^that

^converges

^toapointxin

^{X. Let D} _si

and

ss

^{are as}described intheprevious

paragraph.

Since

13X

^ispretopological

D

has a subnet

H

that convergestox. This means that

H

is compact, contradicting ourhypothesis. Thus Xmustbelocally

compact.

1-1

’We

shall see that the

LC

property will guarantee that

X

isdense in

X f.

We

will nowshowthat,foranycontinuous function f

X -- ^K

from a non-compactHausdorff

LC convergence

space

X

into a compactHausdorff topological

space

K,

X

isaHausdorff compactification ofX.

THEOREM

7. Iff

X K

is a continuous functionfroma non-compactHausdorff

LC convergence space X

into a compact Hausdorff topological

space K

and

X X u S(f)

is equipped with the convergencestructuredescribedabove, then

X

isa compact, Hausdorffconvergence

space

thatdensely containsX.

PROOF. We

willbegin by showing that

X

is compact.Letsbe a universalnetin

X

suchthats is eventuallyinX.

Suppose

sdoesnotconverge^toapointinX.Then the universalnetf*osconvergesto somepointx in

S(f)

(bylemma

3). Hence

sconvergestox in

xf.

Thus

every

universalnetin

X converges

in

X

ObviouslyeveryuniversalnetinS(f) convergesin

X f.

Itfollows that

X

is compact.

To verifythat

X

isHausdorff

suppose

s is anetin

X

that

converges

toboth x andyin

X f.

Ifx

X

thensis

frequently

in

X

and

six

^{convergestox.}Since s has a convergentsubnetin

X

scannotconvergeto apoint yin

S(f);

henceyis in

X.

Since

X

isHausdorff,x y.

Suppose x,y}

S(f).Thismeans that s has no convergent subnet in

X

and thatf*os convergestoboth x and

y

inS(f);hence x y (sinceS(f)is Hausdorff).Thus

X

isHausdorff.

We

will nowshowthat

X

isdensein

X f. Let

x

S(f)

and let

U

beanopen

neighbourhood

ofx in

K.

We

wishtoshowthat there exists anetin

X

thatconvergesto x.

Let M

be an

open

neighbourhoodof x in

K

whoseclosure(in K)iscontained in

U.

Then

clxf--[M]

is non-compact(bylemma4)and so

f-[M]

contains anet with no convergent subnetinX.Since f*ot is anetin

K,

f*ot has a convergent subnet that

converges

tosome point

l(t)

in

S(f) (by

lemma

3). Hence

hasasubnetthatconvergesto

l(t) (by

definition ofthe

convergence

structureon

Xf).

Since

S(f)

^c:

U n

S(f).

Hence

for eachopen neighbourhood

U

of x in

K

there exists anet withno convergent subnetin

X

thatconvergestoapoint l(t)in

U S(f). It

followsthat there is anets

{s 5 A}

of such netsin

X

whose limitsl(s) {l(si)

5 A}

in

S(f)

convergestox.(The open neighbourhoods ofapoint x can be directed by defining

U

^<

U1

^and

^U

^<

U2

^ifU

_ ^U1 U2

^where

^U, U

^and

U2

^{are open}

neighbourhoods of x).

For

each

5 A

let

A

^{denotethedomainof}stiand let_si

{si ^IX

^{e Ai}.}

^We

claimthatthenet

D s" ⁱ

^e

^{A, IX A}

^orderedlexicographically by

A,

thenbyAi, has a subnet that convergestox.

Let T

be a subnetof

D.

Since

X

wasdeclaredtobea

LC space

then

T

has a subnet

H

with no convergent subnet.

We

claim that

H converges

tox.If

U

isanarbitrary open neighbourhood of^{x in}

t

A}

convergesto S(f),then there exists an0te

A

such that {l(si)

i

e

etA} _ ^{U. For 5 otA,} _{si IX

^e

l(si);hence

f*os ^converges

^tol(s).

^Hence

^forany

^{15 otA}

there exists

Ati

^{such that}

{f*os ^IX

t

5

e

ttA, IX IxaAi} U.

Then,for any

5

0tA,

f-[f**s ^IX

^e

^{IxotAi}] f-[U]}

^andso

{s8

_ tx-[U]. Hence

f*oH is

eventually

in

f*otX-[U] U.

Since

U

was anarbitrary

open

neighbourhoodof x f*oH

converges

tox. Since

H

hasno convergentsubnet andf*H

converges

toxthen

H converges

tox (bydefinitionof the

convergence

structureon

xf).

Thismeans thatx e

clxfX

^{and so}

^X

^{isdense in}

^X f.

We

have shown that

X

isaHausdorff compactification of

X.

^l-]

Observethat in the last part of theabove

proof

wehaveshown that, if

X

isa non-compactHausdorff

LC convergence space

andfisa continuous functionfrom

X

intoa compactHausdorff topological space then

X

ispretopologicalateachpointx inS(f).

PROPOSITION

8. If f-

X

--)

K

isa functionfromaHausdorff

convergence space X

intoa compact Hausdorff topological

space K

thenthefunctionf extends continuouslytoa functionf*

X

--)

K

where

f*lsf)

^isthe identityfunction on

S(f).

PROOF.

Clearlyboth

f*ls(0

^and

f*lx

^fare continuous onS(f)and

X

respectively. Letsbe anetin

X

that

converges

toxin

S(f).

Thenf*os

converges

tox

f*(x)

in

S(f) (by

definitionof the

convergence

structureon

Xf). Hence

f*os

converges

to

f*(x).

Thusf*is continuous on

xf.

^i-1

EXAMPLE

9.

Let X

be therealline.

Let

anets"

A -- ^X

^(inX)

^converge

^{toa pointx in}

^X

^ifandonly

if x is anintegerandfor any0te

A

thereexistsa

y >

0tsuchthat

s[yA]

7

(x

1,x]. Observe that

X

isa Hausdorff

convergence space. To

showthat

X

isa

LC space

let

S ss"/i A

beanetofnetseachof

t. t. ie

A, IX

whichhas no convergent subnet in

X. For

eachi

A,

let_si

si ^{I.t AS}}

^andlet

^D

t.

i E, Ix

^e

E8

^{be asubnetof}

^{D. We}

claimthat

T

isnot

A}

(ordered lexicographically).

Let T s8

?> t+l

compact(hence

X

isa

LC space).

If

i

e

;

and

Ix

^{e ;ithenthereexists a}

? ^]

^suchthat

si

(sinceno cofinalsubset ofsi[Ei]isboundedinthe

space

of real numbers

R). Consequently

for each

;

thenet

s= {si ^{Ix Y-i}}

^{has acountablyinfinitesubnet}

^t {si ^Ix

^e

^A}

^{with noboundedinterval}

in

X

containingmorethanfinitely

many

points of

t. Let

^0t

;

and

13 A.

^Thenthere exists a

51 ^{> a}

ⁱⁿ

t

> s ⁺

^1.

Consequently

we canconstructa cofinalsubset

H

of

T

such that

H

Y-.and

Ix1

ⁱⁿ

Ai

^{such that}

si

has no convergent subnet.

It

followsthat

T

isnotcompact; hence

X

isa

LC space.

Let

f"

X

---)[-1,1] bethe functionfrom

X

into

[-1,1]

(equipped withthe usualinterval

topology)

defined as

f(x)

sin(n)ifx (n-l,n] wheren is aninteger. If is anetin

X

that

converges

toapoint

y

(n-l,n] forsomeintegernthen iseventuallyin(n- 1,n]; hence fotis

eventually

sin(n)

f(n). It

then follows thatfis continuouson

X. We

claimthat if

U

isan

open

interval in[-1,1] then there exist an infinitenumber of integersrsuch that sin(r)

U. It

would then follow that

clxff- ^[U]

^{isnotcompactin}

^X

for any

open

neighbourhood

U

in[-1,1].

Let Z

denote thesetof all integers. Ifn

Z

let[ng]denotethe largest integerless than n.

We

will usethe following fact: The set

{n [n]

"n

Z}

isdense (equivalently, uniformly distributed)in[0,1]. (This factis

proved

inmostbooks on numbertheory).

Let

> 0 andmbe

any

number.

We

claimthat thereexistsanintegerr suchthatsin(r) (sin(m) e,sin(m) +

e).

There existsa

5 >

⁰suchthat sin[(m

5,

m

+ i)]

(sin(m) e,sin(m)

+ e). Suppose

m

>

0 and let k

be an eveninteger largerthan m+1. Sincetheset ^ng

[nn]

n

Z

isdense in[0,1 ^{then the}set k(n/

[n]) n

Z

isdense in[0,k]. Then there exists aninteger

Z

such thatk(t: [tn])

[0,k]

andsosin(ktn k[tn])^e (sin(m) e,sin(m)+ e).

But

sin(kt k[tn]) sin(kt:)cos(-k[t/]) + sin(-k[t])cos(ktn) --0

+

sin(-k[tn]), the sine of an integer. Thus if r=-k[tn], sin(r) (sin(m) e,sin(m)+ e). It easilyfollows that

sin-[(sin(m)

e,sin(m) + e)] n

Z

is infinite.

We

arriveatthe same conclusion if wechoosern

<

0.

Hence clxf--[(sin(m)

e,sin(m) +e)] isnon-compactin

X.

Thusf[X]

sin[Z]isdense in

S(f). Hence X

isacompactification of

X

whose outgrowthis

S(f)

[-1,1

].

Theexampleabove illustrates aspecialtype ofcompactificationcalled asingular compactification.

We

define thisbelow.

If thefunctionf

X K

fromaHausdorff

convergence space X

into acompact Hausdorff topological space

K

maps

X

into S(f) then we will saythat fis asingular

function

^{and call}

^X

^asingular

compactification of

X.

Singular compactifications of locallycompactHausdorff

spaces

are discussed extensivelyinAndr6

[1]

and Chandler

[5].

Theyare characterized asbeing those

compactifications ttX

of

X

whose

outgrowth X

isaretractof

ctX.

Thefollowingtheorem follows easily fromProposition 8.

THEOREM

10. If f

X - ^K

^{isasingularfunctionfromaHausdorff}

convergence space X

intoa compact^where

In

example 9

^f*lx

Hausdorff topological^{f and}above, the closed interval

^f’Is(0

^{istheidentity}

space K

^functionthen[-1,1

S(f

^onS(f)^S(f).is aisretractaretractofof

X X

under the

f.

functionf*

X -- ^S(f)

Proposition 11isageneralization of lemma inChandler

[5].

PROPOSITION

11.

Let etX

be aHausdorff compactificationof a

convergence space X

such that 0X’NXis compact.Iff

X--- K

isa continuousfunction from

X

intoa compactHausdorff topological

space ^PROOF. K

that extends^Let

^Y

^clKf[X].tofit

txX ^We --

^are

^K

^thenrequired

^fa[txX]

^toshow

^S(f).

^thatfa[]is contained inclvf[XW]for all

F

Kx. ^{Let F} Kx

^(where

Kx

^isas describedabove).Then^t:tX’g( clax(XW) (since everynetin

F

hasa convergent subnet in

F

and txX’kX

clax(F u XW) claxF ^u claxXkF ^F clax(XW)). Hence

fa[oX’kX]

fa[clax(XW)] clvf[XW].

Since this istrueforall

F Kx, fa[txX’xX]

Kxl

S(f).

Let p KW[otXkX]. Let u

be an

open

neighbourhood (in

K)

^of

p

^{such that}

clKU

^misses fa[]. Then

clfa-[U] fa-[cl,U] X. Hence clxf[U] (= clfa[U])

is acompact subset of

X.

Thisimpliesthatpcannotbelongto

S(f)

(bylemma

4). Hence S(f)

fa[]. !-’1

LEMMA

12.

Let

f

X K

be a continuous function from a Hausdorff

LC convergence

space

X

into a compactHausdorff topological

space K.

Ifix)(isaHausdorff compactification of

X

such that

txX

is compactand f extends continuouslytoff

txX -- ^K

^{sothat faseparates}the points of

txX,

then

txX

is equivalent(asacompactificationof

X)

to

X

^f

X u S(f).

PROOF. By

11,f[txX] S(f).

We

definea function

txX -- ^{X u}

^S(f)asfollows: j(x)

f(x)

ifx belongs to txX’kX andj(x) x ifx belongs toX. Clearly is one-to-one.

We

now verify that is continuous.

Let

s

A X

beanetin

X

such thats

converges

tox inttXkX.

We

wishtoshow that jos j(x)^then(= if(x))^{thereexists an}in

X f.

Equivalently we

^open

neighbourhoodwish

U

toof yshowinthat

K

suchs

--

that

^fa(x) fa(x)

ⁱⁿ

^X

^e

^f. KIU. ^Suppose By

^s8the function

^y

ⁱⁿ

^{X f.}

^If

^y

f

^fa(x) X

K

extends continuouslytoa functionf*

X

^f

-- ^K

^{such that}

^f’Is(0

^isthe identityfunctionon

S(f).

Then f*os f*(y) y

U,

and so there exists a

Ix ^A

^suchthat

^{f*os[llA] U. It}

^followsthat

^s[ktA]

f*[U].

Similarly, since fa is continuous on 0iX, fitos

-

^fa(x); hence there exist a e

A

such that

f%s[SA] KIKU

^and

^s[iA] fa<--[KIKU].

Thisimplies that

tX--[KIK

U]

f-[clKU]

cannot be empty, a contradiction.

Hence y

fit(x). Since s --> y,^s--> fa(x) ^asrequired. Thus is a continuous function.

We

nowproceed similarly^toshow that

j<-

iscontinuous.

Let

s

A

-->

X

be anetin

X

that

converges

to x S(f).

We

wishtoshow that

j-oS

-->

j<--(x) fit-(x).

Equivalentlywe wishtoshow that s-->

<-(x).

Suppose

s---> yⁱⁿ

IxXXX. We

claim thaty

fit*-(x).

If y#

fit<--(x)

thenfit(y)#

fitofit-(x)

x (since fit is one-to-oneonctXXX).

Hence

there exists anopen neighbourhood

U

offa(y) suchthat x

otXXcltxU.

Since fit"0tX

K

is continuous fitoS--> fit(y).

Hence

thereexists a

I.t ^A

^{such that}

fitoS[I.tA] c: U;

then

s[I.tA] _ ^f-[U].

^{Similarly,sincef*}

^X

-->

K

iscontinuousands

--

^xin

^{X f,}

^f*os

^--

^{f*(x) x;hence}

there exists aie

A

suchthat

f*os[iA]

^t:::

KXcl:U.

^Thus

s[SA] _ f*t--[KXclKU]. It

follows that

tX-[KXclaU]

f-[clU]

is non-empty, a contradiction.

Hence y (x)

asclaimed.

It

thenfollows that s.->

fa-(x)

and so

j<-

is continuous. Since 0tX

-+ X

isahomeomorphismthat fixes thepoints of

X, ctX

and

X

areequivalent compactificationsofX. ^[--!

If

G

isa collectionofreal-valued bounded functions on

X,

the evaluationmap

e

^inducedby Gisthe function

e ^X Fl{Ig g

e

G} (where,

for each

g, I

^{isa closedintervalcontaining}

^g[X])

^defined

^by

e(x)

<g(x)>g . ^Note

that the closure in

l-Ig Ig

^{ofe[ X]isa compactset.}

Let

X

be a Hausdorff

LC

convergence spaceand letC*(X)denotethe collection of all real-valued boundedcontinuous functions on

X. We

willshow that, by using ^{the above} method of constructing compactificationsof a Hausdorff

LC convergence space

we

may

constructacompactification X*of

X

in which

X

isC*-embedded,i.e., acompactification X* of

X

whereeveryfunctionfin C*(X) extends continuouslytoareal-valuedfunctionf*onX*. Consider the evaluation

map

ec,tx^induced

by C*(X)

from

X

into

FI{ Ig g C*(X)} (where,

for each

g, Ig

isaclosed boundedintervalcontaining

g[X]).

Then

xec’x X S(ec,o:)).

^Since

^X

^isa

LC space

^andec,x)maps

X

intoa compactHausdorfftopo- logical space,

xec’cx

isaHausdorff compactification^of

X.

Now

ec, cx

extends continuouslyto

ec, cx*

^on

xectx

where

ec,x*

^{restricted to}

S(ec,cx)

isthe identityfunction. If f

C*(X)

and xf"

FI I"

^g

C*(X)} -- ^If

^where

xfoec,cx(x)

^{f(x)thenthemap f* xf*ec,}

tx*

^isa continuous extension offto

xec’tx

mappingapointxin

S(ec,x)

tof*(x)in

If.

^Wehavejustconstructed acompactificationof

X

in which

X

isC*-embeddedand whose

outgrowth

isa compactHausdorff topological

space. We

willdenote

xec’xby X. ^We

^{havepurposelyused a}

^symbol

resembling the oneusedfor the

Stone-ech

compact- ification

13X

^{of alocallycompact}Hausdorff topological space

X

sincethemethod usedto construct

13X

mimicsoneusedto construct

13X

(see 2.2 of Andr6 [1]).

Thefamilyofall Hausdorff compactifications ofaHausdorff

convergence space

canbe partially orderedasfollows:

tzX

<

),X

ifthereexists a continuous functionh

?X

^{0tX from}

?X

^onto0tXsuch

hlx

^{fixesthepointsofX.}

THEOREM

13.

Let X

be aHausdorff

LC convergence space.

Then

13X

^{> 0X for all Hausdorff}

compactifications^oO(of

X

whoseoutgrowth

aX

isacompact Hausdorff topological

space

thatisC*- embeddedin

oX.

Also

X

^>

^?X

^for

^any

compactification

),X

^where

),X

isof the form

xf X S(f)

where f"

X -- ^K

is a continuous functionfrom

X

intoa compactHausdorff topological

space K.

PROOF. Let X

be a non-compactHausdorff

LC convergence

space.

Let oX

be a Hausdorff compactificationof

X

such that

ctX’

is a compacttopological spacethat is C*-embedded in

txX.

Weare requiredtoshow that

ctX

<

13X. ^{Let M}

^f

^C*(0tX):

^fisa continuous extensionto^0t,

X

ofa function in

C*(xXLX)

}.

Since

C*(XLX)

separates the pointsof 0XLX,

M

separatesthe points of 0d.

Hence,

_eM

is one-to-oneon0tXLX. Let

T C*(0X)] _x.

Since each function in

T

extendscontinuouslyto

X,

eT extends continuouslytoa function

ex ^I

^on

X. ^Let

^night

15X

0tX be a function from

ISX

^{to0X which}

maps

eTI3*--(x)

n

15XLX ^toeTa(x)

^{CoXZXfor each x}

^eTa[CXLX]

^{and whichfixesthepointsof}

^X

(notingthat

ewl[15X] ^{eTt[cX]). It}

^iseasilyverified that

x

is continuous.

Hence X

^<

X.

Suppose

that

?X

^{is a}compactification^{of theform}

X X w

S(f) where f

X K

is acontinuous function from

X

into a compactHausdorff topological

space K. Let Y clKf[X].

If

g

e

C*(Y)

then gof*^e

C*(xf).

Since

C*(Y)

separatesthe pointsof

Y

andS(f)

_ ^Y

then the family

{gof* g

C*(Y) separates thepointsofS(f).

Consequently

if

T C*(xf),

eT^{isone-to-oneon}S(f).

Let M TIx.

^{TheneM extends}

continuouslytothe function(eM)13on

X. ^Let lxf 13X -- ^X

be a function from

15X

^onto

^X

^which

maps

(eM)13(x) XXX ^toeT(X)

^{S(f)for eachx}

(eM)I[yXZX]

and which fixes thepointsofX.

Againit iseasilyverifiedthat

nfxf

is continuous.

Hence yX

<

X. l’-’1 EXAMPLE

14.

Let

o1 denote the first uncountable ordinal.

Let X { Ii

[0,ol) where,foreach

[0,01),

Ii

^isthe unit interval[0,1].

We

willsay

thdt

anets in

X

convergestoa rationalnumberx in ifand onlyifs iseventuallyinevery

open

interval in

Ii

^thatcontains x.

A

netsconvergestoanirrational number x in

Ii

^ifandonlyif has an immediatepredecessor and s iseventuallyinevery

open

interval containingxin Ii_1.Thus anets in

Ii

^willalways convergeⁱⁿ

Ii Ii

^/1.

It

iseasilyseen that

X

isa non- compactHausdorff

convergence space. We

will describe someother propertiesof

X.

Weclaimthat

X

isnotpretopological. Let S {si"

5 A}

be anetofnetsin

Ii

^{(i [0,01))where}

eachnetsiconvergestosome irrational numberl(si)in

Ii

^/1-

Suppose

thenetsare chosen so thatthenet {l(si)

i A}

convergestoan irrationalnumber yin

Ii

+ 2.

For

each

5 A,

let

s {si

^{la. Ai} and}

Ix"

5 A, I.t

^Ai} (ordered lexicographically).^Since

D Ii

^{nosubnet of}

^D

^can

^converge

^toa

let

D si

pointin

Ii

+ 2(sinceallnetsin

Ii ^converge

^inIit.)

Ii

^/

^{1). Hence}

^{no subnetof}

^D

^can

^converge

^{toy. Thus}

^X

isnotpretopological.

Also observe thatforthe irrational number

rd4

in some

Ii

^thenets si"

i

A wheresi

rd4

for all i ^Aconvergestothe irrational numbern/4in

Ii

^/1.

Hence

aconstant nets in

X

whereeachmember is the same number r in

X

neednotnecessarily

converge

tor.

We

now claim that

X

is a

LC

space. Let S sti i A be anetofnetseach of which has no convergent subnet in

X. For

each

i A,

letsi

s ^{I.t Ai }.}

^If

^{i A,}

^sihas no convergent subnet Ix"

Zi}

such that in

X

hencenocof’malsubset of

s

^iscontained in

any Ii.

^Thuss/ihas a subnet

t {s ^I.t

Ix"

5 A, :Ei

(ordered lexicographically).

Let T

ti

Ii

^isfinite for each

[0,1). Let D {si

ix.

5 A, I.t As}

(where

A

and

Ai

are cofinal in A and

Ei

respectively).

Let

o

A

and

15 Ai.

^If

s {s ^Ii

then there exists a

51

^>0in

^A

and tlⁱⁿ

Ai

^suchthat

si ^Ii

^/1.

Consequently

wecanconstructa subnet

H

of

T

suchthat

H

hasno convergentsubnet.

It

followsthat

T

is non-compact;hence

X

isa

LC

space.

Letf beanycontinuous function fromXintoa compact Hausdorfftopological space Kand let u be an irrational number in

[1,0]. Let U

{ui"

[0,01)}

whereui uforall

[0,01)

and lets {sti"

5

A be theconstant netinsome

Ii

^{suchthatsi uiforall}

^{5 A.}

^Thenthenets

^converges

^{tothenumberui}

/1. (Note that

U

is

non-compact).

Sincef(s)is aconstant netin

K

andfiscontinuousf(ui+

1) f[s]. It

followseasily that f[U]mustbeasingletonsetin

K {f(u0)}. Let

xbe anarbitrary pointin

f[X]

and let

V

be anopen neighbourhood ofx inclKf[X]. Ifx isan irrational numberthan

clxf[V]

is non-compact (since

clxf-[V]

containsasetsuch as

U

above).

Suppose

xis a rationalnumberinsome

Ii. ^Let

^s si"

5

e A} beanetof irrational numbers in

I,

such that sconverges^tox. Sincefiscontinuousthenetf[s]

converges^to f(x)ⁱⁿclKf[X]. Hence there exists an0e

A

such thatf[s[oA]]^g;V.ThismeansthatV containstheimage ofan irrational number in

Ii.

^{Againit followsthat}

clxf-[V]

isnotcompact.Then,

by

lemma4,ClKf[X]^isthesingularsetS(f)off, i.e.,fisasingularfunction.Sincefisan arbitrary function everyfunctionfrom

X

intoa compactHausdorff topological

space

issingular.

Hence

thecompactification

13X ^xec’

^isasingular compactification (sinceec.(x) issingular).

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Robert

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