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

DepartmentofMathematics

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

tocoincidewiththe 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 beenstudiedand 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-(echcompactification

X

ofa

Ts.s

topologicalspacecan be described asaWallman-typecompactification usingmaxima]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 alarger classof spacesthan do

oX

and[oX. Althoughwehave not yetcharacterizedthe class of spaces forwhich

%X

woX,

wehave shown thatthisclass includesallsubspacesof

R

n.

Thisresultenablesustoshowthat

%X

canexhibit

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}).

ThesetAissaidtobe decregsin

(respectively,

increas/n)

ifA

d(A)

(respectively,

A

i(A)).

Aset whichis eitherincreasingordecreasing 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 spacecanberegardedas atopological orderedspace relative tothediscreteorder

(equality).

Let Ebe the space

[0,1]

with itsusual 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 thatA

]-(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"))

isthefinest

12’-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

Yanincreasing,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)).

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 Fsuch th V

;

equivalently,

(X,

,

)

T-ordered

ifthe order is cloud sub ofXx X. Aspe Xis

T.-odeed

if isfiesthe 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, thereedisjointopen 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 isnottrue,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. Thusy

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 uniquemaximal

CZ-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

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

contradicted,anditfollows that

CZ(%)

isamaximal CZ-filter. Itisobviously the onlymaxims]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 thesetofS]Imaxims]Cg-filtersonX. ByProposition 1.8, these includes]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 ofthispropitionarroutine, 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.

1"21

Let be thetopologyon

)

withclosed subbase

{t

A

CZ(X)}.

Fromthe twopreceding propositions,

itfollowsthat hasanopen subbase of monotone opensets;thus

(),

<

7)

isatopological orderedspace. Let

"7oX

(,

<

).

THEOREM2.5 Forany_{T3 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 issufficient 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 .4has 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 letAbeaCZ-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,andtheproofiscomplete.

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.

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. ByTheorem 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])

aredisjointopenneighborhoods 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 Theorem2.7reducetothestatementthat 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 isthe 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,asinthe 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

NEW ORDERED COMPACTIFICATION

3.

%X

AND

cvoX

The Wallman orderedcompactification

(ooX, o)

ofa

Tl-ordered

spaceXwasintroducedbyChoeand Park

[2]

in1979. Inthis 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 itspropertiescan 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 tobeac-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

isthesetofallmaximal 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’sgeneralization 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

=/oXforanytotally 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 eachclosed,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. IfAisanon-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

PAA1. 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

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 thiscaseisthe 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/$oXcanberepresentedas 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, CompactificationsofOrdered Topological

Spaces,"

Proc. Camb. Phil. Soc. 71

(1972)

463-473.