The Nachbin compactification via convergence ordered spaces

THE NACHBIN

COMPACTIFICATION

VIA

CONVERGENCE

ORDERED SPACES

D.C.KENTandDONGMEI LIU

DepartmentofMathematics

WashingtonState University

Pullman, Vashington99164-3113

(Received April 21, 1992)

ABSTIACT. Weconstructthe Nachbincompactification fora

F3.5-ordered

topologicalordered

space

by tailingaquotient ofanorderedconvergence space compactification. A variationofthis quotient construction leads toacompactification functoronthe category of

F3.5-ordered

convergence ordered spaces.

KEYWORDSANDPHRASES: topological orderedspace,convergence orderedspace,

F2-ordered

space,

F3.s-ordered

space, Nachbincompactification.

1980MATHEMATICS SUBJECT CLASSIFICATION CODE:54D 35,54F05,54A 20

0. INTRODUCTION.

The Nachbin

(or

Stone-Cech-ordered)compactification

(see [1],

[6])

is the largest

T2-ordered

topological ordered compactficaton ofa

Ts.-ordered

topologicalordered space. In

[4],

oneofthe

authors and G.D. Richardsonconstructed an ordered compactification

(X’,

o)

for an arbitrary convergenceorderedspaceX. Thislatter compactification exhibitsessentiallythesame universal

propertyastheNachbincompactification, but behavespoorlyrelativeto separation properties

(see

Example

1.4).

Startkugwithanarbitraryconvergenceorderedspace

X,

weintroduceanequivalencerelation onthe set

IX’I

which underlies

X’,

andobtain anorderedquotient space

X’/R

which

s

both

compact and

T2-ordered.

WenextgivetwoconditionsCand Owhich are necessaryandsuicient tomakethe naturalmapfrom

X

into

X’/

bothanorder embedding andahomeomorphic embed-ding,sothat

X’/R

becomesa

T2-ordered

convergenceordered compactification ofX. Forordered convergencespaces

X

satsfyhugconditionsCand

O,

itturns out that thetopologicalmodification

%X of

X

isa

Ts.-ordered

topological orderedspace,and

%(X’/R)

istheNachbincompactification

of%X.

In

particular, if

X

isassumed to bea

T.s-ordered

topological orderedspace,then

A(X’/)

isthe Nachbincompactification ofX.

regularconvergenceorderedspacesatisfyingconditionsC and 0 to bea

Ts.s-ordered

convergence ordered space, and we show that for such a space

X,

the regular modification

r(X*/)

of the

quotient

X’/

isaregular,

T2-ordered

convergenceorderedcompactification of

X.

Relative to this

compactificationfunctor, the regular, T2-ordered, compact convergence spaces

(with

increasing,

continuousmapsas

morphisms)

formanepireflective subcategoryof thecategory of all

T3.s-ordered

convergence ordered spaces

(with

increasing, continuousmapsas

morphisms).

1. PRELIMINARIES.

We introducesomebasic notation and terminologyand summarize someresults from

[4].

If

(X,

_<)

is aposet, and

A

C_

X,

we denote by

i(A),d(A),

and

A

^

the increasing, decreasing, and

conez

hulls,

respectively,of

A;

notethat

A

n

i(A)

d(A).

_Similarly,if

F(X)

_{is the set of all}

(proper)

filters on

X

and

r

F(X),

let

i(Y’),

the filter generated by

i(F)

F

Y’),

be the increasinghullof

Y’;

thedecreasinghull

d(Y’)

andconvexhull

Y?

aredefinedanalagously.

A

filter

Y"

issaidtobeconvexif

"

f.

Note

that

i(Y’) v

d(Y’).

If

(X,

<_,

-,)

is a

poser

(X,

<)

equipped withaconvergence structure--,_{which islocally convex}

(i.e.,

f

z whenever

Y"

--,

z),

_then

(X, <,--,)

_iscalled aconvergence orderedspace; weusually

write

X

rather than

(X, <, -,)

when thereis nodanger of ambiguity.

A

convergence ordered space

is

Tx

-ordered if the sets

i(x)

and

d(x)

areclosed for allx

X,

and

Tn-ordered

if the order

<_

is a

closed

subsetof

X

X. Foranyconvergence ordered space

X,

let

CI(X)

(respectively,

CD’(X))

denote thesetofall continuous, increasing

(respectively,

decreasing)maps from

X

into

[0,1].

A convergenceorderedspace whoseconvergencestructureis atopology iscalled a topological orderedspace. Suchaspace is said tobeconvexif the open monotone

(i.e.,

increasingor

decreasing)

setsformasubbase for the topology. Fortheremainderofthis paper, weshall adoptthe notational abbreviation used in

[4]

andwrite

"t.o.s"

instead of topological ordered

space"

and "c.o.s." in

placeof

"convergence

orderedspace".

A t.o.s. X is said to be

T3.a-ordered

if it satisfies the followingconditions:

(1)

Ifz E

X,A

is a closed subset of

X,

and z

A,

then there is

y

CI’(X)

and g

CD’(X)

such that

f(z)

g(z)

0 and

f(y)

V

g(y)

1, for ally

A;

(2)

Ifz y in

X,

there is

f

CI’(X)

such

that

f(y)

0and

f(z)

1. The

Ts.s-ordered

spaces are preciselythosewhich allow

T-ordered

t.o.s, compactifications, andall

Ts.-ordered

spacesareconvex.

IfX is a

T.-ordered

t.o.s., thenthe Nachbin compactification ofX

(see

[1],

[6])

is obtained

by embedding

X

intoan "orderedcube

,

whose componentintervals are indexedby

CI’(X).

The

Nachbincompactification/0Xischaracterizedby the followingwell-known result.

PROPOSITION 1.1. If

X

is a

T3.-ordered

t.o.s., then

0X

is

T-ordered.

Furthermore, if

f

X

--

Y

is an increasing, continuous mapandY is a compact,

T-ordered

t.o.s., then

f

has a unique,increasing,continuous extension

f’/0X

-,_Y.

Wenext describe briefly the constructionofthe convergence ordered compactification

X’

of

an arbitrary c.o.s.

X

described in

[4],

which has essentially the same liftingproperty as

0X.

Given a c.o.s.

X,

let

X

be the set of all non-convergentmaximal convexfilters on

X,

and let X’

{

z

X}

X

.

Before proceeding

further,

it will be useful toestablish the following proposition about maximalconvexfilters.

PROPOSITION1.2. Themaximal convex filters on aposet

X

are precisely theset

{f

Y"

is anultrafilter on

X}.

convexset (7

e

,

the filters

’1

and

r2

generated by

{i((7)f

F" FE

r}

and

{dC(7

respectively,arewell-defined filters finerthan,and hence equalto,

r.

Thus

i((7)

implies

i(G)

n

d(G)

G

;

therefore

.

f.

Again assuming that Xis anarbitrary c.o.s., let

o

X X’ be defined by

(x)

5, forall x C X. A partial order

<_

is defined on X as follows:

"

_<

iff

i(3

r)

<_

(or,

equivalently,

d(.)

<_

r).

Since x

_<

y iff

<_

,

’(X, <_)

(X

,

<_)

is anorderembedding.

IfAC_

X,

letA’

()"

EX AC

Y’);

if

Y"

C

F(X),

let r _{denotethe filter in}

F(X)

_generated by

(F’

FC

Y’).

Aconvergence structure on

(X

,

<_)

isdefinedasfollows: For C

F(X),

*-,

Eo(X)

itfthere is

Y

-,_{zsuchthat Y"_<4;}

Writing

X’

inplace of

(X’,

_<’,

-,),

westate the following resultwhich is provedin

[4].

PROPOSITION1.3. If

X

is ac.o.s., then

(X’,

)

is aconvergence ordered compactification of X. If

f

X

Y

isacontinuous, increasingmapand

Y

acompact, regular,

T2-ordered

c.o.s., then

f

hasaunique, increasing, continuous extensionf,

X"

--

I/’.

Recall that aconvergencespace

Y

is regular if

clr

"

xwhenever x. Here "c/r" is the closure operator for

Y,

and

citY:

isthefilteron

Y

generated by

(clyF" F

’).

In

[4],

ac.o.s. Xisdefined to be strongly

T2-ordered

if

X

is

T

(i.e.,

convergentfiltershaveunique

limits)

and the followingconditions hold:

($1)

if

Y"

z,.

X

e,

and

i(’) _<

.,

then

d(.)

_<

;

(S)

if

Y

-

,

.

X’,

and

d(Y’)

_<

.,

then

i(.) <_

.

In

Proposition 2.8,

[4],

it isshown that

X"

is

T2-ordered

iff

X

isstrongly

T2-ordered.

As we see inthenext example,very nice c.o.s.’smay fail tobestrongly

T2-ordered.

EXAMPLE

1.4. Let

X

be the Euclidean planewith its usual

(product)

order and topology. Let

Y

be the filteronXgenerated bysets of the form

F

{(a, b)

X

-

<

a

<

0, b

0}

for

each natural number n, and let x

(0,

0).

Let

.

betheconvexhull ofany ultrafilter containing

theset S

((a, b)

EX a -b

-1)

andcoarser than thefilter generated by sets of the form H,

((a,b)

_

X" b

>_ n)

forn 1,2,3,.... Note that

($1)

is violated by

3r,.

and x; thus the compactificationX"ofXisnot

T2-ordered.

2.

0X

AS A QUOTIENTOFX’.

Let

(X,

<_

be any c.o.s., and let

(X,o)

be the convergence ordered compactification of

X constructed in the last section. By Proposition 1.3 thereis, for any

f

CI(X),

a unique,

continuous,increasingextension

fo

X"--,

[0,1].

Wedefineanequivalence relation onX" as follows:

{(r, 9)

X" X"

f,(Y)

f.(.),

for all

f

CI’(X)}.

Let a be the projection map ofX" onto

X’/

(i.e.,

for each

Y

E

X’,

o(’)

[Y’],

where

[Y’]

is the-equivalenceclass containing

Y).

Apartial order

_<

on

X’/]

isdefinedasfollows:

[’]

_

[]

iff

f.(’)

_

f.()

inRfor all

f

e

CI’(X).

We also impose on

X/]

the quotient convergencestructure which is described

(see [2])

as

follows: If

F(X/)

and

[Y’]

C

X/],

then -,

[Y’]

_in

X/]

_iffthereis

Y

C

[Y’]

andthere is afilter {E

F(X)

such that {

*-

Y

in

X

and

({)

_<

.

PROOF.

X’/,

is obviouslycompact. Toshow that

X’/P.

is T-ordered, it is sufficient

(by

Proposition1.2,

[4])

toshowthatif

,

(9_E

F(X’/),

[F]

and(9 --,

[]

in

X’/,,

and

hasatraceontheorder

_,

then

[r]

<:

[.].

If

f

e

CI’(X),

define

f

X’/

[0,1]

by

/([3r])

f.(Y’),

for all

,v

X’. It is easy to verify that

f

iswell-definedand

f

e

CI’(X’/).

If

[Y’]

and O

[.]

in

X’/,

and

hasa traceon

_<,

itfollowsthat

f()

f(O)

hasa trace on theorder of

[0,1];

since

[0,1]

is

T2-ordered,

f([’])

f,(.T)

_

f,()

f([.]).

The latter inequality holds for all

f e CI’(X),

andso

[,v]

_

[],

which establishesthat

X’/.

is

T2-ordered.

Foranarbitraryc.o.s.

X,

wehave alreadydefined thecontinuous,increasing maps

o

X

--,_X’ anda X"

X’/;

we define

o

X

X’/

by

o

ao

o.

It is clear that

o(X)

is dense

inthe compact,

T2-ordered

c.o.s.

X’/.

Weare nowinterested in characterizing those spaces X

for which

(X’/.,,)

is acompactification. With this goal in mind,we introduce the following conditions.

CONDITIONC. Forany maximalconvexfilter

Y"

onX,

r

x inX

iff/()

.f(x)

in

[0,1]

for all

f

CI(X).

CONDITION O.Foranypoints x,y inX,x

_<

y inXiff

f(x)

<_ f(y)

in

[0,1],

forall

f

CI’(X).

It is easy to verifythat any

Ta.-ordered

t.o.s, _{satisfies ConditionsC and O.}

LEMMA

2.2. ffXis ac.o.s, satisfying ConditionsC andO,then

[]

(k),

for allx X. PROOF.

CI*(X)

separatespoints in

X

byCondition

O,

andso a isone-to-oneon

(X).

This

implies

[]

if x y.

Next,

assumethat there is

Y"

X

[].

Then

f.(Y’)

f,()

f(x)

for all

f

CI(X);

inother words,

f(,)

f(x)

in

R,

forall

f

CI*(X).

ConditionC then implies

:T

x in

X,

contradicting theassumption

:T

X’.

THEOREM 2.3. Let

X

be a c.o.s. Then

X

--,

X’/

_{is an order and a homeomorphic} embedding iff

X

satisfiesConditionsC andO.

PROOF. Suppose that

X

satisfies Conditions C and O. Then is one-to-one since

CI’(X)

separates pointsin

X.

Also note that

a.o

(a]{x))

oo, and thus

al(x

isone-to-one. Let --.

[3]

in

X’/,.

Then there is { E

F(X’)

such that {

-**

in

X"

and By definition of convergence in

X"

there is a filter

Y"

on

X

such that

"

x and

Therefore,

o1()

_>

l(a())

_>

ol(cr(.T’))

-.

(a[(x))-(a(Y")).

It follows by Lemma

2.2 that

(al,(x})-(a(.T))

>_

.T*.

Consequently,

o1(@)

_>

o-’(Y")

r

__,

x

o1([]),

i.e.

()

-.

([]).

Thu i

Let

[]

_

[]

in

X/;

then for any

f

E

CI(X), f(]c)

_

fo(),

i.e..f,(o(x)) <_ f(o(y)),

whichimplies

,f(x) <_

f(y),

for all

f

e

CI(X).

By Condition

O,

z

<_

y. Thus

o

is increasing, andweconcludethat

o

is anorderand homeomorphicembedding.

Conversely,assumethat

o

isbothanorderandhomeomorphic embedding.Let

Y"

beamaximal convexfilteron

X

suchthat, for some

e

X,

]’()

]’(x)

forall

f

e

CI’(X).

Suppose

nottrue. Thenweneed to consider twocases.

CASE 1.

Y"

y andy

-

x. Thisimplies that foreach

f

CI*(X),

y(Y’)

y(y).

Fromthis wededucethat

[]

[],

whichisacontradiction,sincep is assumed to be one-to-one.

CASE 2.

Y

X

.

Thisleadstothe conclusionthat

[Y’]

[3];

inother words,

p(Y)

[3]

in

X/,

which implies

Y"

x in

X,

since is ahomeomorphic embedding. This contradicts

Y

E

X’.

Wetherefore conclude thatXsatisfies ConditionC.

follows since is anorderembedding. Therefore,

X

satisfies Condition O.

THEOREM2.4. Forevery c.o.s.

X

satisfyingConditionsC andO,

((X/), )

isa

T2-ordered

c.o.s, compactification ofX. Furthermore,for anycompact, regular,

T2-ordered

c.o.s.

Y

and for

anycontinuous, increasing map

f

X

Y,

there is a unique, continuous, increasing extension

fp.

:X’/

r.

PROOF.Thefirst assertion is animmediate corollary of Theorem2.3. Thesecondfollows easily

withthehelp ofProposition 1.3.

For any c.o.s.

X,

let

c#oX

be thet.o.s, consistingofthe

poser

(X,

<)

with the weaktopology

inducedby

GI’(X).

Notethat

GI’(X)

PROPOSITION2.5. Let

X

beac.o.s, satisfying ConditionG. Let

X

oX

betheidentity map. Then is anorderisomorphismandahomeomorphismrelativetoultrafilter convergence.

PROOF. It is obvious that is a continuousorder isomorphism. Let

"

--. x in 0X, where

F

is an ultrafilter. By Proposition 1.2, is a maximal convexfilter and

f(F)

-

f(z)

implies

f(])

f(x)

in

[0,1],

for all

f

E

CI’(X).

Condition C thus guarantees that

-

x in

X,

and

hence

r

zinX.

PROPOSITION2.6. IfXisac.o.s, satisfying ConditionsGand

O,

then

c#oX

is a

T3.s-ordered

t,o.s.

PROOF. Firstobserve that

coX

also satisfies Condition C and O; O is obvious, and C fol-lowsfrom Proposition 2.5,since andC#oXhave the sameultrafilter convergence andhence, by

Proposition 1.2, thesameconvergenceof maximalconvexfilters.

For / E

C’I*(oX),

let !be the closed interval

[0,1]

indexed by /, and let P

C(X)

be equipped with the usualproductorder andproduct topology. Then Pis acompact,

T-ordered

t.o.s. Define

o

CoX

P by

o(X)

,

where

:

C(C#oX)

[0,1]

is givenby

(/)

/(x),

for all /

C’I*(C#oX).

SinceC#oX has the weak topology induced by

CI*(CoX)

C*(),

and

C*(o)

separates pointsinC#oX by Condition O,

o

is atopological embedding

(see

8.12,

[10]).

By Condition

O,

o

isalsoanorderembedding.

Givena c.o.s.

X

satisfying C and

O,

weintroduce someadditional functional notation. Let

betheevaluationembedding of the

T3.s-ordered

t.o.s,

c#oX

into itsNachbincompactification and let e

eo-i

X

-/o@oX).

The unique extensionofe to X’

(guaranteed

by Proposition

1.3)

is denoted by e., and theextension ofe to

X’/P.

(guaranteed

by Theorem

2.4)

is denoted by

e.

If

f

Gf’(X)

GI’(oX),

the unique extensions of

f

in

Cf’(X’)

and

GI’(/o(oX))

(see

Proposition 1.3 and

2.4)

aredenoted by

f,

and f’, respectively. The followingcommutative

diagramishelpfulinkeeping track of thesevarious maps.

x

x.

x./

oX

o

THEOREM2.?. IfX is anyc.o.s, satisfyingG’ and

O,

then

e

isanorder isomorphism anda

homeomorphismrelative to ultrafilter convergence.

POO.

s

[Zl

IS;l

i

x’/

itr

.(z)

.()

t

t([Zl)

X*/

sincethelatterspace iscompact. Itfollows by uniqueness offilter limits inbothspacesand

the continuityof

e

that

el(a)

a.

If

X

is any convergence space, let

AX

denoteitstopological

modification (i.e.,

X

isthe set

IXI

equippedwiththe finesttopological structurecoarserthan

X.)

If

X

is a c.o.s, satisfying C and

O,

weobtainfromProposition 2.5and Theorem2.7 that

X

oX

and

(X’/))

is acompact,

T2-ordered

t.o.s, _{homeomorphic and order isomorphic under}

e

to

o(taoX).

LetOo

woX

--

X/

bedefinedbyOo o

o

o -1

o

o -1.

COROLLARY2.8. If

X

is a c.o.s, satisfying C and

O,

then

(A(X’/R),Oo)

is the Nachbin compactification of

caoX

,X. If

X

is a

T3.5-ordered

t.o.s., then

((X’/R),o,)

is the Nachbin compactification-ofX.

Onequestion which deserves clarificationisthe status of

X’/

as a"quotient" of

X’.

Wehave

indeedequipped

X’/R

withthe quotient convergencestructure, butcan weinterpret

_

as the "quotient order" relative to theorder_’ definedon

X’?

Various notionsof "quotientorder" have been considered

(for

instance,see

[5]

and

[8]),

butthe order

_

isgenerally different than these. Insteadofregarding the order andconvergencestructuresof

X’/

separately,wethink thatit is

appropriate to consider thenotionofa "quotient

c.o.s.",

where orderand convergence structures

areconsideredtogether. Fromthisperspective,thenext theorem indicatesthat

X’/R

isindeeda quotientc.o.s, of

X’,

atleastinthe category of

c.o.s.’s

which satisfy ConditionsCandO.

THEOREM2.10. Fora c.o.s.

X,

let

X"

and

X’/R

bedefinedasbefore. Let

Y

beany c.o.s.

satisfying CandO,andleth"

X’/]

-

Y. Thenhiscontinuousand increasing iffho

X"

Y

iscontinuousandincreasing.

PROOF. If hiscontinuousand increasing, thesame isobviouslytrue for ho

.

Conversely, suppose ho# is continuousandincreasing. Let q

--

[Y’]

in

X’/R;

then there is

’

E

[Y’]

andafilter/ on

X"

such that/{ yt inX" and

_

r(A).

Henceho

r(A)

---, ho

(.7

)

in

Y,

bycontinuity ofhor. But @

_

r(A)

and

r(

r)

[r],

so

h(@)

--,

h([.]),

_implyingthat his continuous.

To showthat h is increasing, let ey be the natural map from Y into

o(woY)

and consider g

er

oho o

o

X

--

o(taoY).

_{Since g}

woX

_--*

o(caoY)

is alsocontinuous and increasing, thereis acontinuous,increasingextensiong"

o(taoX)

(wY)

whichmakesthe diagram below commute.

x

x.

x’lA

Y

o(woX)

--,

o(Y)

g.

Thus

erohoo

g"oeoo, dsince

o

X

X’/

isadense iection,

eoh

g"

o.

But

er

order embedding,soh

e

og"o_{e, d h} increg.

3.

T3.s-ORDERED

CONVERGENCE ORDERED SPACES.

In

this briefconcluding section, weintroducethenotion ofa

Ta.s-ordered

c.o.s., describe the largestregular,

T2-ordered

c.o.s, compactification of sucha space, and interpret this

compactifi-cation inthe languageofcategorytheory. The necessary categorical terminologycanbe foundin

corn-pactification. In

[9],

it is shown thatthe Hausdorff,completely regular convergencespaces, which weshall referto as

T3.s

consergence spaces areprecisely thoseconvergencespaceswhichallow a

regular, Hausdorffconvergencespacecompactiflcation.

Given a convergence space

X,

let

rX

denote the regular

modification

of

X

(i.e.,

rX

is theset

IX

equipped with the finestregular convergencestructurecoarserthan theoriginal convergence structureon

Wedefineac.o.s.

X

which isregularandsatisfies conditionsCand O to bea

T3.s-ordered

c.o.s..

Itfollowsby Proposition2.5thata

T3.s-ordered

c.o.s.

X

has thesameultrafilter convergenceasits

topological modification

AX

woX.

THEOREM3.1. Let

X

be a

T.s-ordered

e.o.s, andlet

r/oX

r(X/R)

be theregular

mod-ificationof

X’/].

Then

(roX,)

is aregular,

T-ordered

c.o.s, compactification ofX. If

Y

is aregular, T-ordered, compactc.o.s, and

f

X

--,

Y

_{is continuousand increasing, then}

f

has a unique,continuous, increasingextension

,

:roX

Y.

PROOF. By Theorem 2.3,

X

--.

X’/]

is an order embedding and a homeomorphic embedding. By the functorial propertiesofthe regularmodification and the fact that

rX

X,

it follows that

X

--,

roX

_{is continuous. Because}

X’/

_and

roX

have thesame ultrafilter convergence,it iseasy to verify that theregular modification of

p(X) (considered

as asubspace of

X’/)

coincideswith

o(X)

consideredas asubspace of

roX.

Fromthiswe seethat

x

isalso continuous,and the first assertion is established. Thesecond assertion isanimmediate consequence

of Theorem2.4.

We denoteby C the category of all

Ts.s

ordered

c.o.s.’s,

with increasing continuous maps as morphisms; letDbethe fullsubcategoryofCconsisting of allregular, compact,

T-ordered

c.o.s.’s. If D C’isthe inclusionfunctor,itfollows byTheorem3.1thatthe functor

ro

C

D,

which assigns to eachobject

X

inCits compactification

roX

and toeach morphism]" X Yin Cthe extension

fo

roX

THEOREM3.2. If C andD arethecategories defined inthe precedingparagraph,thenD is anepireflectivesubcategory of C.

IfX is a

T.s-ordered

t.o.s.,it isgenerallynot true that

oX

roX,

althoughit is true inthis casethat

oX

(roX).

The

T3.s

convergence spaces mentioned earlier in this section are the

T3.-ordered

c.o.s.’s for whichthe partialorderisequality. Indeed,any

T.s

convergence

space’X,

equippedwiththe trivial order

(equality),

satisfies ConditionCandOrelativeto

CI’(X)

C’(X),

theset ofall continuous mapsfromXinto

[0,1].

Forsuchaspace

X,

oX

(which

also hasthe trivial

order)

coincides with thelargest regular,Hausdorff convergence space compactificationofXconstructed in

[9].

REFERENCES

[1

P.Fletcher andW.Lindgren, Qui-Uni/orm Spaces, Lect. Notesin

Pure

and Appl. Math., Vol. 77,Marcel

Dekker,

Inc.,

New

York

(1982).

[2]

D.C.

Kent,

"Convergence

Quotient

Maps",

Fund. Math. 65

(1969)

197-205.

[3]

D.C.Kent andG.D. Richardson,"Completely Regular and c0-RegularSpaces", Proc. Amer. Math. 8oc. 8:

(1981),

649-652.

[5]

S.D.McCartan,

"A

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(1968),

317-322.

[6]

L. Nachbin, Topology and

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Van Nostrand,

Nero

York Math. Studies, 4, Princeton, N.J.

(1965).

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G.

Preuss,

Theorl

o

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V.

ReidelPubl.

Co.,

Dordrecht

(1987).

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H.A. Priestley, "Ordered Topological Spaces and the Representation of Distributive

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($)

114

(1972),

507-530.

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G.D.Richardson andD.C.

Kent,

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,

Proc. Amer. Math. 8o. 31

(1972),

571-573.