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 O}which 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.

Inadditionto givinganalternateconstructionfor the Nachbin compactification,weobtainsome interestingresultspertaining to convergence orderedcompactifications. In Section 3,wedefine a

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 a}poset, 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 a}convergence orderedspace; weusually write

X

rather than

(X, <, -,)

^{when thereis no}danger 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}.

PROOF.Clearly every maximalconvexfilter istheconvexhull of everyfinerultrafilter. Con- versely,suppose

Y"

is anultrafilteron

X

and is a convexfilter suchthat

f ^<_ ..

^{Thenfor any}

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 denote}the 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 a}convergence ordered compactification of X. If

f ^{X Y}

^isacontinuous, increasingmapand

Y

acompact, regular,

T2-ordered

c.o.s., then

f

^hasRecall that^aunique, increasing, continuous extensionaconvergencespace

Y

is regular iff,

clr X" --

^I/’.

"

^{xwhenever x.} Here "c/r" is the

closure operator for

Y,

and

citY:

^{isthefilteron}

Y

generated by

(clyF" F ’).

In [4],

^{ac.o.s. Xis}defined 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 of}any 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

^{({) _<} .

THEOREM2.1. Forany c.o.s.

X, X/]

^{is acompact,}

T2-ordered

^c.o.s.

PROOF.

X’/,

^{is obviouslycompact. Toshow that}

X’/P.

^is T-ordered, it is sufficient

(by

Proposition1.2,

[4])

^toshowthatif

,

⁽⁹E

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 now}interested in characterizing those spaces X for which

(X’/.,,)

^{is a}compactification. 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 anorder}and homeomorphicembedding.

Conversely,assumethat

o

^{isbothanorderand}homeomorphic 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. This}implies 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.

Finally,let x,y Xsuchthat

f(x) < y(y)

forall

f CI’(X).

^Then

fo(p(x)) <_ f((y))

for all

f

^G

CI’(X), i.e.f.() _< f.()

^{for all}

f CI’(X).

This implies

[3] ^<_ []

in

X’/,

^{and x}

<_

y

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

bea^c.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. IfXisa^c.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#oX^{have the same}ultrafilter 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 a}topological 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 itsNachbin}compactification 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 any^c.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)

one-to-one. Furthermore,

(X)

^{is dense}

in/o(oX),

which implies that the extension

e

^{is onto}

/o@oX).

^{It followsfromTheorem 2.4 that}

e

is continuous and increasing. Finally, if is an

X*/

^{sincethelatterspace iscompact. It}follows 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

quotient^c.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. Thenhiscontinuous}and increasing iffh^o

X"

Y

iscontinuousandincreasing.

PROOF. If hiscontinuousand increasing, thesame isobviouslytrue for h^o

.

Conversely, suppose h^o# 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 ofh^or. 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 also}continuous 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"oe, 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

In [3],

^aconvergencespace

X

is definedto be completelyregular ifitallowsasymmetric 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 thesame}ultrafilter convergenceasits topological modification

AX woX.

THEOREM3.1. Let

X

be a

T.s-ordered

^{e.o.s, andlet}

r/oX r(X/R)

^{be the}regular 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 continuous}and 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 a}subspace of

X’/)

coincideswith

o(X)

^{consideredas a}subspace 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 to}each 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 of}all continuous mapsfromXinto

[0,1].

^Forsuchaspace

X, oX (which

^{also has}the 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 and}G.D. Richardson,"Completely Regular and c0-RegularSpaces", Proc. Amer.

Math. 8oc. 8:

(1981),

^649-652.

[4 A

Compactification forConvergenceOrderedSpaces",Canad. Mah.

Bull. 27’

(1984)

^505-512.

[5]

S.D.McCartan,

"A

Quotient OrderedSpace", Proc. Camb. Phil. Soc. 64

(1968),

^317-322.

[6]

L. Nachbin, Topology and

Order,

Van Nostrand,

Nero

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

(1965).

[7]

^G.

Preuss, Theorl o

Topological8tructures,

V.

ReidelPubl.

Co.,

Dordrecht

(1987).

[8]

^H.A. Priestley, "Ordered Topological Spaces and the Representation of Distributive Lat- tices", Proe. LondonMath. 8oc.

($)

114

(1972),

^507-530.

[9]

^G.D.Richardson andD.C.

Kent,

"RegularCompactificationofConvergence Spaces

,

^Proc.

Amer. Math. 8o. 31

(1972),

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[10]

S. Willard, General Topology, Addison-Wesley Publ.

Co.,

Reading,

Mass. (1970).