Compactifying a convergence space with functions

(1)

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

(2)

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

istxX’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 sconvergestox.

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 andonlyiffor 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).

(3)

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 thatye

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-compact_Hausdorff

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 thefunction 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 anynetofnetsin

X

suchthateachnet

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.

(4)

Suppose

that,for eachie

A,

l(t)isthe limit ofsome convergent subnet ti

={si

IX

Ei} ofsi. Since

13X%X

iscompact thenet {l(ti)

5

A}

hasa subnet{l(ti)

5

E}

whichconvergestosomepointxin it. it"

5

_e

Y.,

tx

Z5}

(itself a subnet ofD).Then

T

isof the form

T

{si

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

thatconvergestox

(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 asdescribed 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

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

sdoesnotconvergetoapointinX.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

(5)

neighbourhoods of x).

For

each

5

A

let

A

denotethedomainofstiand letsi

{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 ofx 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,

letsi

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

(6)

be an eveninteger largerthan m+1. Sincetheset ng

[nn]

n

Z

isdense in[0,1 then theset 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 compactHausdorff topological

space K

then

S(f

is aretractof

X

under thefunctionf*

X

--

S(f)

where

f*lx

f and

f’Is(0

istheidentityfunctiononS(f).

In

example 9above, the closed interval[-1,1 S(f)isaretractof

X

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

space

K

that extendstofit

txX

--

K

then

fa[txX]

S(f).

PROOF.

Let

Y

clKf[X].

We

arerequiredtoshowthatfa[]is contained inclvf[XW]for all

F

Kx.

Let F

Kx

(where

Kx

isas describedabove).Thent:tX’g( _{clax(XW) (since every}netin

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

space K.

Ifix)(isaHausdorff compactification of

X

such that

txX

is compactand f extends continuouslytoff

txX

--

K

sothat faseparatesthe 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)(= if(x))in

X

f.

Equivalently wewishtoshowthats

--

fa(x)

in

X

f.

Suppose

s

y

in

X

f.

If

y

fa(x)

thenthereexists an

open

neighbourhood

U

of yin

K

suchthat

fa(x)

e

KIU.

By

8the functionf

X

K

extends continuouslytoa functionf*

X

f

--

K

such that

f’Is(0

isthe identityfunctionon

S(f).

Then

(7)

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 similarlytoshow 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,txinduced

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 compactHausdorff

topo-logical space,

xec’cx

isaHausdorff compactificationof

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 alocallycompactHausdorff 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 compactificationsoO(of

X

whoseoutgrowth

aX

isacompact Hausdorff topological

space

thatis C*-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

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

(8)

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 acompactificationof theform

X

X w

S(f) where f

X

K

is acontinuous function from

X

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),

eTisone-to-oneonS(f).

Let

M

TIx.

Then_{eM 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

/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

(9)

e A} beanetof irrational numbers in

I,

such that sconvergestox. Sincefiscontinuousthenetf[s] convergesto 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

space

issingular.

Hence

thecompactification

13X

xec’

isasingular compactification (sinceec.(x) issingular).

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Robert

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Richard

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