A NEW ORDERED COMPACTIFICATION

Internat. J. ^Math. & Math. Scl.

VOL. 16 NO. (1993) 117-124

A NEW ORDERED COMPACTIFICATION

D.C.KENT

DepartmentofPureandAppliedMathematics Washington State University

Pullman, WA99164

T.A.RICHMOND Department^ofMathematics WesternKentucky University

BowlingGreen, KY42101

(Received September I0, 1991 and in revised form April II, 1992)

ABSTRACT. A new Wallman-type ordered compactification

%X

^is constructed using maximal C

Z-

filters

{which

^havefilterbasesobtainedfrom increasing and decreasingzero

sets)

^asthe underlyingset. A necessaryand sufficient conditionisgiven for

oX

^{tocoincidewith}the Nachbin compactification

[oX;

ⁱⁿ particular

%X oX

^whenever

^X

^{has thediscreteorder. The} Wallman ordered compactification equals

oX

whenev.er XisasubspaceofR". Itis shownthat

oX

^isalways

^T,

^{butcanfail to be}

Tx-ordered

or

T2.

KEY WORDS AND PHRASES.

CZ-set,

maximalCZ-filter,

T-ordered

^space,

T2-ordered

^{space,Nachbin}

compactification, Wallman ordered compactification.

1991AMSSUBJECTCLASSIFICATION CODES. 54F05, 54D35,^54D10

0. INTRODUCTION.

L.Nachbin

[10]

initiatedthestudy of ordered compactifications whenhe characterizedthe topological ordered spacesthatallow

T2-ordered

compactifications

(we

^callthese

Ts.s-ordered spaces),

and constructed the largest such

T-ordered

compactification [oX by embedding X in an

ordered

^{cube. The Nachbin}

(or

^Stone-(ech

ordered)

compactifieation

oX

^{has beenstudied}and applied byvarious authors

(see,

for instance,ouroddnumbered

references).

^Asecond ordered

(but

^notnecessarily

T2-ordered)

compactiflcation

oX,

cared the Wdlman ordered eompactification,^wasintroducedby Choeand Park

[2].

^Anecessary and sufficient conditionfor

woX

^{[oX was givenin}

[6],

^{and in}

[8]

the separation propertiesof

woX

^were

investigated.

Itis well known

(e.g.,

see

[4])

^{thattheStone-(ech}compactification

X

^ofa

^Ts.s

topologicalspacecan be described asaWallman-typecompactification using^maxima]filtersofzerosetsasthe underlyingset for thecompactification. Wehaveextended this construction to

Ts.s-ordered

spaces, and the result isa newordered compactificationwhichwecall

%X.

^Thisnewcompactification,like[oXand

woX,

has the universal extensionproperty for increasing, continuous mapsintocompact,

T-ordered

^{spaces. Withthe} helpofthis universalproperty,weobtainnecessary andsufficientconditions for

%X

[oX;inparticular thisequality holds when the order ofXisdiscrete. Asanalternativeapproachtoconstructing

oX, %X

ismoresatisfactorythan

oX,

in thesensethat

oX

^and

floX

^{coincideon a}larger classof spacesthan do

oX

^and[oX. Althoughwehave not yetcharacterizedthe class of spaces forwhich

%X ^woX,

^wehave

shown thatthisclass includesallsubspacesof

R n.

Thisresultenablesustoshowthat

%X

^canexhibit

thesame "pathological behaviorrelative toseparation properties thatwasdemonstrated for

oX

ⁱⁿ

[8].

118 D.C. KENT AND T.A. RICHMOND Forexample,

%X

^{fails tobe}

T-ordered

^{ifX R for,t}

^>

^3.

Itremainsanopen question whether[oXcanbedescribed viaaWallman-typeordered compactification forall

Ts.s-ordered

^spaces

X’.

1. PRELIMINARIES.

Let

(X, <)

^bea

poser

and letAbeanon-empty subset ofX. Let

d(A) {z

EX"z

<

aforsome aE

A)

and

i(A) {z

EX"a

<

zforsomeaE

A),

incaseA

{z},

^wewrite

d(z)

and

i(z)

rather than

d({z))

^and

i({z}).

^{ThesetAissaidto}be decregsin

(respectively, increas/n)

^ifA

d(A) (respectively,

A

i(A)).

Aset whichis eitherincreasing^ordecreasing issaid tobemonotone;if

A d(A)Oi{A),

^thenAisconvez.

If

f" {X, <) (Y, <)

^isafunction betweentwo

posers,

then

1’

^is

increas/n9 (respectively, decrea/ng)

^if

z

<

yinXimplies

f(z) ^< f(I/) (respectively, (I/) ^< (z)

inY.

A topologicalordered space

(X, <,r)

isatriple consistingofaposet

(X, <)

^{and aconvex topology}

r on

X;

ris conveziftheopen monotone setsformanopensubbase. Theterm space willalwaysmean topological orderedspace,and

(X, <, r)

^{willbeshortenedto}

X

when thereisnoambiguity.Notethat every topological space^canberegardedas atopological orderedspace relative tothediscreteorder

(equality).

Let Ebe the space

[0,1]

^{with its}usual order andtopology. For anarbitrary space X we denote by

CI*(X) (respectively, CD*(X))

^theset of all increasing

{respectively, decreasing),

^{continuous maps} from

X

intoE. An

increa.sin

^zeroset

(respectively, decreasin9

^zero

set)

^isasetof the form

f-l(0)

where

f CD*(X)

(respectively,

f ^CI*(X)).

^Thesetof all increasingzerosets

(respectively,

decreasingzero

sets)

^onXwillbe designatedby

IZ(X) (respectively, DZ(X)).

^Usingstandardproceduresdescribedin

[4],

^oneeasilyproves the next twopropositions.

PROPOSITION1.1 IfXisaspace,

f _ CI*{X), CD*(X),

^andaE

E,

then:

DZ(X); (b) f-([a, 1]) IZ{X); (c) g-([0,a]) IZ(X),

and

(d) 9-([a, _1]) DZ(X).

(ffi) /- ([o, ,,l) ^e

PROPOSITION 1.2 Foranyspace

X, IZ(X)

^and

DZ(X)

^areclosed undercountable intersections andfinite unions.

AsubsetAofaspaceXiscalledaC-zeroset

(or C2"-set)

if there isB

I2"(X)

^andC

DZ(X)

^such

thatA B cC. Let

CZ(X)

^bethesetof all CZ-setsonX. One easilyverifiesthe following.

PROPOSITION1.3

(a)

^A

CZ(X)

^{iffthereis g}

CI*(X)

^andh

OD*(X)

^{such that}A

]-(0),

where

1/2(g

-t-

h). (b)

^Theset

C2"(X)

is closed undercountable intersections.

It isgenerally not true that

C2"(X)

^isclosed underfiniteunions; forinstance,

[0,1]

^and

[2,3]

^are

CZ-setsin

R

whose union is notaC2"-set.

By a

falter :?"

^on

X,

^wealways mean aproper setfilter

(one

^{that does not contain}

).

Thefilter on X generated by

(z),

forz

X,

will bedenoted by 9. Ifafilter 1"has afilter base ofincreasing zerosets, then }" is called an 12"-filter;

D2"-fdter

^and

C2"-filter

^{aredefinedsimilarly. For an arbitrary}

filter }" on

X,

let

I2"(}’) (respectively, D2"(}’), C2’(}’))

be the filter on

X

generated by }"

I2"(X) (respectively,

^}"

c DZ(X), Y nC2"(X)).

Notethat

IZ(t)

(respectively,

DZ(I), C2"(1"))

^isthefinest12’- filter(respectively, D2"-filter,

CZ-filter)

^{coarserthan}’.} Thenextproposition follows fromZorn’s Lemma.

PROPOSITION1.4 If )"isaC2"-filter

(respectively,

12"-filter,

2"-filter),

thereissmaximalC2"-filter

(respectively,

I2"-filter,

D2"-filter)

^{finer than}’.}

PROPOSITION1.5 Let

X,

Ybespacesand[

X

Y^anincreasing,continuous map.

(a)

^IfA

IZ(Y)

(respectively, A

DZ(Y),

^A

CZ(Y)),

^then

I-(A) I2"(X)

(respectively,

I-(A) DZ(X), .f-(A) . ^CZ(X)).

(b)

^{If}" isafilter on}

^X,

^then

IZ(I(}’)) < 1(I2"(}’)), D2"(f(}’)) _< I(DZ(Yf)),

and

CZ(I(}’)) < I(CZ(I)).

PROOF.

(a)

IfA6

IZ(Y),

^thenA

g-’(0),

forg

CD*(Y).

^Then

J’-’(A) (9 I)-’(0),

^where

g

CD*(X),

^d

/-(A) IZ(X).

^{The ohercs esimilar.}

(b)

followseilyfrom

().

A spe X is defined

o

be

T-odeed

if, for eh

X, i(z)

d

d(z)

^{re cled}

se.

Aspe X

T-odeed

if, whenever z Fⁱⁿ

X,

^there n increing neighborhood of z nd dreing neighborhood V of F^{such th} V

;

equivalently,

(X, , ) T-ordered

ifthe order is cloud sub ofXx X. Aspe Xis

T.-odeed

^{if isfies}the followingconditions:

(1)

^Ifx

^X,

^Aisa

cloud subt of

X,

andz

A,

then thereis

/ CI*(X)

^d

CD*(X)

^such

that/(z) g(z)

0d

/(F) v g(F)

^for

A; (2)

^If F in

X,

thereis

/ CI*(X)

such

that/(F)

⁰

and/(z)

^1.

The

Ts.s-ordered

^spiese prisely thesubspesof compt,

T-ordered

^sp

(see [10]).

Aspe

X

is defined to be

T-ordered

ifit is

T-ordered

d, whenever A d

B

e disjoint cld subts with Adreing d

B

increing, there^edisjointopen ts and

V,

the forr dreing,

t

latter increing, suchthatA HdB V. Notethat: compt d

T-ordered T-ordered Ts.s-ordered T-ordered T-order.

^obrvethat

T-order T, T-ordered T,

d

Ts.s-order Ts.s (i.e.,

^{completelyregular d}

T);

^{it isnot}true,however,that

T-ordered T.

h the reminder of thksection we

exane

some properties of

Ts.s-ordered

^{spies, th spi} emphisontherole played by CZ-sets.

PROPOSITION 1.6 Let

X

be a

T.s-ordered

^spe. t z

X,

d let

(z)

^{be the filter of}

nighborhoods ofz.

() () CZ(())

() () o ^{o o m o (X ) (X B), CZ(X) a DZ(X).}

() CZ(X) a ,u, o

X.

(d)

^{ff isafilronXsuchthat} z,then

CZ() .

PROOF.

(a)

t Vbe anopenneighborhood ofz. Then thereare

CI*(X)

d g

CD*(X)

such that

() g(z)

0d

I(F) v g(F)

_{if F} X V. Then

-([0, ]) ^g-[(0, }])

^{is CZ-set}

neighborhood of which subset ofV.

{b)

^Let l,g,and V be h theproof

of(a).

IfB

1-’{1)

^{d A}

g-’{1),

^then A

DZ{X),

B

IZ(X),

^andz

{X- A) {X- B)

^V.

(c)

^d

{d)

follow iediately from

{b)

^d

{a),

respectively.

PROPOSITION1.7

In

a

Ts.s-ordered

^spe

X,

^thefollongstamenareequivalent:

(a)

^{z y;}

(b) Z() ; (=) OZ() .

PROOF. It obvious that

{a) (b).

^Toshow

{b) (a),

suppose y is in eh member of

IZ(X)

contningz, ^butz y. Then thereis

f CI*(X) su

that

f(y)

0d

f{z)

^{1. Thus}y

f-{1),

but

f-l(1)

^{isamember of}

IZ{X)

^contningz. Thestablishesthat

{a) {b),

^d

{c) {a)

^followsby adu

arment.

Inthe next sectionweshl constructacomptcation bdonmaxalCZ-filters. Thenext two propositionswillbeuseful inthisendeavor.

PROPOSITION1.8 IfXisa

Ts.5-ordered

^{space andz}

X,

the

CZ()

^isthe uniquemaximalCZ- filteronXcoarserthan

.

PROOF. Wealready know that

CZ(%)

^isthefinest CZ-filtercoarserthan

.

^{Suppose isaCZ-filter}

and

CZ()

<

.

^{ThenthereisaCZ-set G such thatz X G. By} Proposition

1.6(a),

^thereisa

CZ-neighborhood Hof zsuch thatH X-G. SinceH

CZ(),

the assumption that

CZ()

<

,

^is

120 D.C. KENT AND T.A. RICHMOND

contradicted,anditfollows that

CZ(%)

isamaximal CZ-filter. Itisobviously the only^{maxims]CZ-filter} coarserthan k.

PROPOSITION1.9 Let

f ^{X Y}

^beacontinuous,increasing map, where

X

is

Ts.s-ordered

^and

^Y

iscompact and

T2-ordered.

^{If is amaximalCg-filteron}

X,

there isauniquepointI/.u E

Y

such that

f()

IJ inY.

PROOF. Let

Y"

be an ultrs]ilteron

X

such that

,q _< Y’.

^Since

Y

is compact andT2, there is aunique pointI/J in

Y

such that

f(Y’)

I/s. Because isamaximal Cg-filter,

Cg(Y’) <_ t,

and

f(),l) ^>_ f(CZ(Y’)) ^>_ CZ(f(Y’))

^followsby Proposition 1.5.

But f(Y’)

I/ implies

CZ(f())

lttby Proposition

l.e(d),

and therefore

2.

THE

COMPACTIFICATION

Throughoutthissection,we assumethat

X

is

of

X

isapair consisting ofacompactspace

Y

andamap

X Y

such that isbothatopological andanorderembeddingof

X

into

Y

suchthat

tr(X)

^isdense in

Y. In

thissection,weshall constructan ordered compactification

(IoX, b)

of

X

and establishsomeofits basicproperties.

Let be thesetof^{S]Imaxims]Cg-filters}onX. ByProposition 1.8, these include^s]Ifiltersof the form

CZ(),

where zEX. Arelation and

DZ() ^{< J.}

PROPOSITION2.1

(, <

^{is a}

^poser.

PROOF. It isclear that

and

DZ(N) ^_< .

^{Since is aCZ-filter,}

^{IZ(.t/) v DZ(N) .q,}

^{and so /}

^{<_ J.}

^Itis alto true that

IZ(.q) <_

^.t/and

IZ(N) ^_<

PROPOSITION2.2 z

_<

_Iin

X

if

CZ(%) < CZ(I)

ⁱⁿ

^}.

PROOF. Ifz

_< ,

thenbyProposition 1.7,

IZ(%) IZ(CZ(%)) ^<_ ,

which implies

IZ(CZ(%)) ^<_

CZ(I).

Likewise,

DZ() <_ ,

which implies

DZ(CZ()) <_ CZ(%).

Thus

CZ(%) < ^CZ(I).

This reasoning isreversible.

Foranarbitrary,non-emptysubset

A

of

X,

wedefine

{ J }" A J}.

PROPOSITION2.S Let

A,B . ^CZ(X).

() .n tn’-.

(b) u = ^,u".

()

(d) .. ^{Z(x), ,.n i}

^ni.,..i.

^,

ⁱⁿ

^,.

PROOF. All of the rtions ofthispropition^arroutine, andwe and

,

^then

ZZ() <_

and is anincreasing set.

w. nx

^d,- ,/,

b ,/,(.) CZ(*),

for

,

X. Sy Propiion

.Z,

,/,i.

.

^order

.embedding of

X

inX. Weomitthe routineproofofthenextproposition.

POPOSTO Z.

()

For=y

c_ X, ,/,-() ^c ,.

(b)

^If

, _CZ(X),

h.n

,/,-() ,

-d

,/,-(X- ) X ..

1"21

Let be thetopology^on

)

withclosed subbase

{t

A

CZ(X)}.

^Fromthe twopreceding propositions, itfollowsthat hasanopen subbase of monotone opensets;^thus

(), < 7)

^isatopological orderedspace.

Let

"7oX (, < ).

THEOREM2.5 ForanyT3 s-ordered^spaceX,

(7oX, )

^isanordered compactification forXwhose topologyis

Tx.

PROOF. First notethat

"

^X

^%X

^isatopological embeddingby Propositions

1.6(c)

^and

2.4(b);

isalsoanorderembedding,as weobserved previously.

Toshow that

oX

^{iscompact,it is}sufficient to show that any collection

C {i A CZ(X),i I}

of subbasic closed sets in

%X

^with the finite intersection property has a non-empty intersection. If

. ^{{A I},}

^{then .4}has the finite intersectionproperty by Proposition

2.3(a).

Let 31be any maximal CZ-filtercontaining

q;

then31

Toshow that

7Xo

^is

^Tx,

^let)4,91/be two distinct maximal CZ-filterson X. Then therearedisjoint CZ-setsM 31 andN )/. ItfollowsthatX Nisaneighborhoodof31 not containingI/,andX M isaneighborhood of)/notcontaining 31.

Finally,if31

),

^then

(31)

converges to 31in%X,andtherefore

(X)

^isdensein

oX.

The next theorem shows that

%X

^hasthesameuniversal extensionproperty as

woX

and

BoX.

THEOREM2.6 LetXbea

Ta.5-ordered

^{space,Yacompact,}

T:-ordered

space, and

f

^{X Y be}

acontinuous,increasingmap. Thenthereisaunique continuous, increasingmap

]" _"loX

Ysuchthat the diagrambelow commutes.

X

PROOF. Let

]’loX Y

be definedby

](3t) ,

^where is defined in Proposition 1.9. We firstshow that

]

^isincreasing. Let 31

<

^{) in}

^2;

^then

DZ(.Y) ^<_

31.

Suppose y) y inY. Then there is g

CI*(Y)

^{such that}

g(y.)

^and

g(y)

0. Thus y;

g-([0, ]) ^DZ(I(M)),

^since

^f()/)

^{y inY. But}

g-([i,1]) I(31),

^since

I(31)

Y, and therefore

)’(31) DZ(I()I)). However, DZ()I)

<_ 31 implies

DZ(f())) ^<_ f(DZ()I)) ^_< )’(31)

follows by Proposition 1.5. This contradiction establishesthaty

_<

y;, andso

]

isincreasing.

Wenextshow that

]

is continuous. Let 31

"oX

^{and letAbea}CZ-neighbgrhood ofy.u inY. Fromthe factthat

)’(31)

y.,wededuce that31

)’-x("A),

^andit is easy toseethat

](f-(A))

^{C_ A. Itremains to}

show that

)’- (A)

^isaneighborhoodof31in

"loX.

^Forthis purpose,weemploy Proposition

1.6(b)

^toobtain

C

DZ(Y)

^{and D}

IZ(Y)

such that y

(Y C)

ⁿ

(Y D) _

^{A. Since}

^{(Y C)}

ⁿ

^{(Y D)}

itfollowsthat31

() f-x(’C)) () f-X(D)).

The latter set is open in

"7oX

^andasubsetof

This establishes that

f-x(’-’A)

^isaneighborhoodof31which maps intoA,^{andtheproofis}complete.

THEOREM2.7 Let Xbe

Ta.5-ordered.

^Then

oX BoX

^{iff thefollowingconditionshold:}

(1)

^IfM

IZ(X),

^N

CZ(X),

^{andM N} 0,thenthere ish

CI*(X)

^{such that}

h(N)

^0and

h(M)

^1.

(2)

IfM

DZ(X),

^N

CZ(X),

^{andM N}

,

^{thenthere ish}

CD*(X)

^{such that}

h(M)

^0and

h(N)

^1.

PROOF. Since

foX

^{is the largest}

T2-ordered

compactification of

X,

Theorem2.6 implies that

’7oX

^,6oXiff

%X

^is

T-ordered.

^Thusthe proofwillbe achievedby showing that the specifiedconditions arenecessaryandsufficient inorder for

%X

^tobe

T2-ordered.

122 D.C. KENT AND T.A. RICHMOND

Assume that

"loX

^is

Tz-ordered

^{and let M and N beas} indicated in

(1).

^By Proposition

2.3(a),

f

,

^{and and are bothclosedsubsets of}

7oX.

Furthermore,

.r

^{isincreasing in}

"fox

^by

Proposition

2.3(d).

^Let

d()

^{denote thedecreasinghull of in}

7oX.

^Then

d()

^isclosedbyProposition 4, page44,

[10],

^and

d(r)

ⁿ

^2lr

^{I/t. By}Theorem 1, page30,

[10],

there is g in

CI*{"IoX)

^{such that}

g(l)

0 if4E

d()

^and

g()

ift

E/f.

Settingh gok weobtain

(1).

^Asimilarargumentestablishes

(2).

Conversely, assumethe twoconditidns, andlet

t,

beelementsof

"loX

^{such that}

t . ^M.

^Then

either

IZ()

^{/ or}

DZ(/) .

^If

^IZ()

^/,then

^(because

^{/ is amaximal}

^CZ-filter)

^thereis

M

IZ()

^{andaCZ-set N / such thatM n N}

.

^{If his asstated in}

^(1),

^then

^{h-l([0, ))}

^and

h-l((1/2, 1])

^{aredisjointopen}neighborhoods of^/and respectively, the former decreasing and the latter increasing. If

DZ(/) ,

^wecanapply

(2)

to achieve thesameresult.

1

If

X

has thediscreteorder,conditions

(1)

^and

(2)

^{of Theorem2.7reduceto}thestatementthat disjoint zero setsin X are "completely separated" inthesenseof

[4].

^{Sincethis is truefor any}

Ts.s

^{space, we}

conclude that

%X

^[oX

fix

^wheneverXisa

Ts.s-ordered

^{space withthediscreteorder.}

Asweshall seein thenext section, therearesimple examples of

Ts.s-ordered

^{spacesfor which}

7oX

is not

T2-ordered.

^{In thiscase,wemay beinterestedtoknowwhen}

%X

^satisfiesthe weaker separation properties

"T2

^or

"Tl-ordered’.

This sectionconcludeswith twotheorems pertaining tothisproblem.

Examples showing that

%X

^neednotsatisfy these latterseparationaxiomsarealso provided in thenext section.

THEOREM2.8 LetXbea

T3.s-ordered

^{space. Then}

%X

^is

T2

^iff,for each ultrafilter

Y"

on

X,

there isauniquemaximal CZ-filter onXsuch that

CZ(Y’) ^<_ .

PROOF. Assume

%X

^is

T2

^{and let}

"

^beanultrafilteronX. Then

(Y’)

converges tosome %X, where isamaximal CZ-filteronX. Itmust betrue that

CZ(Y’) _< ;

otherwise { and

CZ(Y)

^would

containdisjoint CZ-setsMandA, andX-A would be aneighborhood of in

%X

^{notbelongingto}

(Y’).

If therewereanothermaximal CZ-filter/finerthan

CZ(Y’),

^then

(Y’)

^wouldalso converge to / in%X,contradicting the assumption that

%X

^is

T2.

^{Thus is}the uniquemaximal CZ-filtersuch that

cz() ^<_ ..

Conversely,assumethat

%X

^isnot

^T;

^{thenthereisafilter on}

%X

^convergingto distinctelements and / in

%X.

^Let

Y"

beanultralilteronXcontaining thefilterbase

{A

^_C X ^/

}.

^{One easily}

verifiesthat

(Y’)

converges toboth and in

%X.

^{Thisimplies,asin}the preceding paragraph, that and /arebothmaximal CZ-filters finerthan

CZ(Yf),

^whichcontradictsthe uniquenesscondition.

THEOREM2.9 LetXbea

Ts.5-ordered

space suchthat,foreach A

e CZ(X), i(A) e IZ(X)

^and

d(A) DZ(X).

^Then

%X

^is

T-ordered.

PROOF. ForS C_%X,let

iv(S

denote the increasinghullof S and

clS

^theclosure of Sin

%X.

^We

willshowthatforarbitrary %X,that

cl(i()) i(),

^andhence

i()

^{isclosed in}

%X.

^The

dual argumentestablishesthat

d{

^{isalsoclosed.}

First,observe thatif /

cl(i()),

^{then for each A}

CZ(X)

^{such that /}

X- A,

there is

i()

^{such that}

X"-

^A. Inotherwords,if

//cl(i(t)),

then foreach A

CZ(X)

such that A

,

^thereis /%Xsuch that

<

^andA

.

Let

cl(i()).

If

/ i({),

^{then /,andsoeither}

IZ()

/or

DZ(.V) .

^Asumethe

former;then thereisM

,

f

IZ(X)

^{such that}

M

/. But /

cl(i({))

implies thereis

such that M

.

^However

^<

^implies

^{IZ() _< ,}

^acontradiction. Ontheotherhand, suppose

DZ(/)

^{. Since ^)d isamaximal CZ-filter, there isaCZ-setM ^{ and N /

DZ(X)

^such

thatMfN

,

^{andhence N c}

i(M) .

^Butby assumption,

i(M) IZ(X),

^andso

i(M) IZ().

Again, /E

cl(i(())

implies thereis

>

^{such that}

i(M)

f[

t.

However

i(M) IZ({) <_ I

isagain acontradiction. Wetherefore conclude that

i()

isclosedin

%X.

NEW ORDERED COMPACTIFICATION

3.

%X

^AND

cvoX

The Wallman orderedcompactification

(ooX, o)

^ofa

Tl-ordered

spaceXwasintroducedbyChoeand Park

[2]

^{in1979. In}this sectionwefind conditionsunderwhich

%X cvoX;

^thisleadstoexamples showing that

"oX

^canfail, in various ways,topreserve theseparation propertiesT2, T2-ordered,^and

T1-ordered.

The construction of

ooX

^{andadiscussionof its}properties^can be foundin

[8]. ^Here,

^wereviewonly

afew relevant facts. Although

%X

^canbe defined forany

Tl-ordered

^space

X,

weshall assume, asin thepreceding section, ^that XisTa.s-ordered,since it isonly for such spaces that

oX

^and

c#oX

^canbe compared.

IfAis anynon-empty subsetofX,let

I(A)

denote thesmallestclosed,increasingset containingAand

D(A)

^{thesmallestclosed,decreasingoversetofA. Ais said tobea}c-set ifA

I(A)n D(A).

^AspaceX

iscalledac-spaceif, forevery c-setAC_

X, i(A) I(A)

^and

d(A) D(A).

AfilteronXwithabase of c-sets iscalled ac-filter. The underlyingsetfor

ooX

^{isthesetofall}maximal c-filterson X. Indeed,the constructionsof

cvoX

^and

7oX

^areverysimilar,withthec-setsplaying thesameroleinthe former that the CZ-setsplayinthe latter. Inparticular, if everyc-set in

X

isa

CZ-set,

then

ooX %X.

^{Thus the}

following propositionisobvious.

PROPOSITION3.1 Ifevery increasing closedsetinXis in

IZ(X)

^andeverydecreasing closedset inXisin

DZ(X),

^then

oX

"/oX.

Another usefulfact,provedin

[6],

isthe following.

PROPOSITION3.2 AspaceXhas the property that

ooX oX

^iffXisa

T4-ordered

c-space.

THEOREM3.3 If

X

isa

T4-ordered

^{spacesuch that,forany sets}

F,

Gin

CZ(X), I(F)

^G

implies

I(F) D(G)

^anddually, then

%X floX.

PROOF. Weshowthat,underthe given assumptions,Xsatisfies conditions

(1)

^and

(2)

^ofTheorem 2.7. To verify

(1),

^{let ME}

IZ(X)

^{and N e}

C2(A)

^{be disjoint.} Since

I(M) M,

^it followsby our assumptionthatM

D(G) .

^{Thuswe can applyNachbin’s}generalization ofUrysohn’sLemma

(see

Theorem 1,page30,

[10])

^{to obtain}

e CI*(X)

^suchthat

f(M)

^1and

/(D(G))

^O. This establishes condition

(1);

the proof of

(2)

is similar.

I

COROLLARY3.4 IfXisa

T3.s-ordered

space such that

cooX

=/oX,then

ooX oX.

PROOF. If

ooX oX,

then,by Proposition 3.2,Xisa

T4-ordered c-spa,e.

^{Everysuchspaceclearly}

satisfiesthe requirementsofTheorem 3.3, andsotheconclusionfollows.

1

A

T-ordered

space whose underlyingpartialorder isatotal

(or linear)

order is calledatotallyordered space. Itis shownin

[7]

^that

ooX

=/oX^foranytotally orderedspaceX.

COROLLARY3.5 IfXisatotally orderedspace, then

ooX %X

^[oX.

THEOREM3.6 LetXbeasubspaceofR

PROOF. InviewofProposition 3.1,it is sufficienttoshowthateachclosed,decreasing subsetofX is in

DZ(X)

^{and each}closed,increasingsubset ofXis in

IZ(X).

Webegin by defining

(in

the terminology of

[3])

^aquasi-pseudo-metricpon

X

defined asbllows: If :r=

(Xl,’--,Xn),

^y=

(Yl,’’’,Yn),

^then

p(x,y)= (Yl- xl)

^V0

+.-.+ (Yn- xn)

^{V0. IfAis}anon-empty, closed, decreasing subset of

X,

wedefine PA X

[0, oo)

^{as follows:}

pA(X)

inf{p(l/,x) y

A}.

Finally, let

hA ^X

^Ebedefinedby

hA

PA^A1. Itfollows that

hA CI*(X)

^and

hl(0)

^{A. Thus}

A

DZ(X).

The dual argument shows thatanyclosed,increasing subset ofXis in

IZ(X).

ItisshowninTheorem3.4 of

[8]

^that

ooR oR

^iffn

^<_

2;^thisyieldsthefollowingconsequence of Theorem3.6.

124 D.C. KENT AND T.A. RICHMOND

COROLLARY3.7

%R" oR"

^iffn

^_<

^2.

Werecalltwoexamples from

[8]

involving subspacesofR in which

ooX,

and hence also%X,fail to exhibit basicseparation properties. LetS

{(z,I/)

^-1

^_<

^z

^_<

1,-1

_<

y

_< 1}

^{beasubspaceofR}

^2.

^In

Example3.6of

[8],

thesubspace

Xl

^S

{(0, 0))

of

R

has the property that

%Xl

^isneither

T-ordered

nor

T2.

^{InExample3.7of}

[8],

^thesubspace

X

^S

((0, I/)

-1

_<

I/-< 1and t/

0)

has the property that

%X2

^is

T

^butnot

T-ordered.

^Wedonotknow ofaspaceXfor which

%X

^isT]-ordered butnot

T.

Asafinalexample, recall thatifXisa

Ts.s-ordered

space withthe discreteorder,then

%X

^{[$oX. If,in} ddition,

X

ischosennottobeT4,then

ooX (which

^{in this}caseisthe ordinary Wallman

compactification)

fails tobeT2,and consequently

ooX %X.

4. UNSOLVEDPROBLEMS.

Find necessaryandsufficient conditionson aspaceXfor

%X

^tobe

T-ordered.

(2)

Find conditionsona spaceXwhicharenecessary and sufficient for

%X woX.

(3)

^{Determinewhether}

%R

^sis

T.

(4)

^Finda

Ts.s-ordered

^{spaceXforwhich}

ooX, oX,

and

%X

^aremutuallynon-equivalent.

,(5)

Determinewhether/$oX^canberepresentedas aWallman-typeordered compactification.

REFERENCES

[1]

^J.Blatter, "Order Compactification of Totally OrderedSpaces," J. Approz. Theory13

(1975)

^56-65.

[2]

^{T. H.} Choe and S. Park, Wallman’s

Type

Order Compact’fication,"

Pacific

^{J. Math. 82}

(1979)

339-347.

[3]

P. Fletcher and W. Lindgren, Q,

asi-Uniform

Spaces, Lecture NotesinPureand Applied Mathemat- ics, Vol. 77,MarcelDekker,

Inc.,

NewYork

(1982).

[4]

^{L.GillmanandM.Jerison, Rings}

of

^{ContinaoesFnctions,}Van Nostrand,Princeton

(1960).

[5]

G.Hommel,"IncreasingRadonMeasuresonLocally Compact OrderedSpaces," Rendiconti Mathe- matica9

(1976)

^85-117.

[6]

^D.C.

^Kent,

^Onthe Wallman OrderCompactification,"

Pacific

^{J. Math. 118}

(1985)

^159-163.

[7]

^{D. Kent and T.} Richmond, "Ordered Compactlflcation ^ofTotally Ordered Spaces," lnternat. J.

Math. l Math. ^$ci. 11

(1988)683-694.

[8]

,"Separation Properties of the Wallman Ordered Compactification," ^lnternat. J.

Math. Math. Sci. 13

(1990)

^209-222.

[9]

T. McCallion, Compactifications^ofOrdered Topological

Spaces,"

Proc. Camb. Phil. Soc. 71

(1972)

463-473.

[I0]

L. Nachbin, TopologyandOrder,VanNostrand, New^YorkMath. Studies, 4, Princeton, N.J.

(1905).