ORDERED COMPACTIFICATION OF TOTALLY ORDERED SPACES

ORDERED COMPACTIFICATION OF TOTALLY ORDERED SPACES

D.C.KENT

Department

ofPureand Applied Mathematics WashingtonState University

Pullman, Washington 99164-2930 and

T.A. RICHMOND Departmentof Mathematics

Western

Kentucky University BowlingGreen,Kentucky42101 (Received January 15, 1988)

ABSTRACT. A complete description of the

T2-ordered

compactifications ofatotally ordered space

X

is given in termsof the simple and essential singularities.

KEY

WORDSAND PHRASES. Totallyorderedspace,

Trordered

compactification, simplesin- gularity,essentialsingularity, Wallman ordered compactification

1980MATHEMATICAL SUBJECTCLASSIFICATION CODE. 54

D

35,54

F

05.

O. Introduction.

A

totally ordered space

X

is defined to bea totally ordered set equippedwith a topology whichis locallyconvexand

T-ordered (i.e.,

^{the orderis a}closed subset of

X X).

^A study of ordered compactificationswasmadepreviouslyby J. Blatter

[1].

^{Ourgoalisto giveamoreintu-}

itivetreatmentofthissubject basedonthenotionof "singularity,"which istheterm thatwe use to designate anon-convergent,monotone, free, convexfilter

(or,

equivalently, anon-convergent maximal

c-filter).

The singularities ofatotallyorderedspace

X

maybeclassified inseveral ways

(e.g.,

bounded or

unbounded,

^increasingor

decreasing),

but the most important distinctionis between simple

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

and essential singularities. For every

T2-ordered

compactificationof

X,

there is a uniquecorn- pactification point corresponding to eachsimplesingularity;itfollows thatatotallyordered space which has only simple singularities hasaunique

T2-ordered

compactification. Essentialsingulari- tiesalwaysoccur asordered pairs, andto a given

Trordered

compactificationeach pair ofessential singularitiescontributes either one ortwo compctification points. There is

(as

Blatter showed

earlier)

^{a smallest}

T2-ordered

compactification obtainedby assigning a single compactification point to each pair ofessential singularities, and a largest

T-ordered

compactificationobtained by assigning two compactification points to each essential pair. The latter is, of course, the Nachbin

(or ^Stone-(ech ordered)

compactification. Anyother

Trordered

compactificationmay be described by partitioning the set

p(X)

^ofall essential pairs of singularities^{into two}subsets, and assigningonecompactification point to each pairinthefirstsubset and two compactification pointstoeach pair in the second. Thus thereis anatural one-to-onecorrespondencebetween the subsets of

p(X)

andthe

Trordered

compactificationsof

X.

Every T2-ordered

compactification ofatotally orderedspace is also totallyordered, and the compactification space always has theordertopology.

1.

Totall/

^Ordered Spaces.

We shallassume throughout this paper that

X

is a totally orderedset. If a, b are distinct elementsin

X

anda

_

^z

_

^{b impliesz a orz b,then b issaid to cover a.}

^A

^{subset A of}

^X

is increosing

(respectively, decreasing)

^ifa E

A

anda

_

^z

(respectively,

^z

_ ^a)

^{implies z E A.}

For

a

e X,

let

[a,---)

^be the set of upper boundsof a, and let

(a,--) {x e ^X

^a

< x}

^be

the proper upper boundsof a; the sets

(-, a]

and

(--, a)

^are defined dually.

For

a,b

X,

we definethe

"open"

interval

(a,b) (a,--), (--,b)

^andthe "closed" interval

[a,b] [a, )f(,b].

If

A

C__

X,

let

i(A) U{[a,--,)

^a

A)

denote the increasing hull of

A, d(A)

^{the decreasing}

hull of

A,

and

A ^ i(A) n d(A)

the convezhull of

A.

If

A A^,

then

A

is calleda convezset.

Let A (respectively, A t)

^{designatetheset}of all upper

(respectively, lower)

^{bounds of}

^A.

We

shall always usethe term "filter" to meanaproper set filter.

A

filter ^jris

.free

^{if thereis}

no1)ointcommon toall thesets in

Y’. A

filterwhichisnot freeis]ized: in varticular,the^symbol

will denote thefixedultrafiltergenerated byapoint

For any filter jr, let jr^ be the filter generatedby

{F ^ ^F

^E jr}; if ^jr jr^, then ^jr issaid to be convex. The setof upperbounds ofafilter jr is definedto be ^jrr

t2{Fr ^F

^E

Y’};

^{y’t is}

defineddually. Foranyfreeconvexfilter ^jr on

X,

the sets jrT and jrt partition X.

A

filter jr is said to be increxsing (respectively,

decreasing)

^{if jrt 6} jr(respectively,^{jr E} jr); ^a filter which is either increasingordecreasingis saidtobe monotone.

Thefree,^convexfiltersareofthreedifferenttypes,which may be describedas follows.

Proposition 1.1 Eachfree,convexfilterjron atotally ordered set

X

is of exactlyoneof the following types:

(a)

Increasing, in which case ^jr has a filterbase consisting of sets of the form

(x, 4)

^fjrt,

where

(b)

Decreasing, in whichcase jr has afilter base consistingof sets of the form

(--,x)

wherex jr?;

(c) Non-monotone,

in which case^jrhasafilter base of sets ofthe form

(a, b),

^{wherea rtand}

b jr?.

In

thiscasejr

n,

where

(generated

^{bysetsoftheform}

(x,-)njrt,x e jrt)

is increasingand

(generated

bysetsof the form

(*-,x) ^)rr,

^z

e r)

^isdecreasing.

If

.

^{is anyfree} ultrafilteron

X,

one may easilyverify that

.^

^{is afree,} convex filter. Since the sets

.r

^and

.1

^partition

^X,

^{at least one of these sets is in}

.,

^{andhence in}

.^.

^{Thus the}

convexhull ofanyfreeultrafiltermustbe monotone. Withthe help of theseobservationsand the preceding proposition,we obtainthe next proposition.

Proposition 1. A flee, convex filter jr ismonotoneiff there is a free ultrafilter

.

^onXsuch

that ^jr

.^.

Furthermoreif ^jr is a monotone, flee, convex filter and )/ is any ultrafilterfiner thanjr, then^jr 4^.

The order topology0 on

X

has as anopen subbase all sets of the form

(a,-,)

^and

(-,b),

^for

a,b X. Thistopologyis locallyconvex

(meaning

that the neighborhood filter at eachpointhas afilterbase ofconvex

sets)

^and

T2oordered (meaning

that theorderrelationisclosed in

X

x

X).

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

Weshallwrite

-r ⁰ x"

to indicatethatafilter

:T

converges toapoint x in theorder topologyon

X. Itiswellknown thatfor anytotally ordered set,orderconvergence coincides with convergence inthe order topology;this canbestatedasfollows: 0__{xiffx sup inf}

Proposition 1.3 Every free,^convexfilter

"

^on

^X

^{isof exactlyone}of the three following forms.

(a) rt

or

.v ,

^{but not}both;

(b) r

xforsomexE

X;

(c) ’t #

^and

r = ,

^butsup

^’t

^andinf

^rT

^{bothfail to exist.}

Proof.

^{Ifa freeconvexfilter}

"

^hasthe property

’t ,

^then

rt X. In

particular, and

’

^{cannot bothhold.}

^Next,

^supposethat

^(a)

^{doesnothold,sothat}

^r?

^{and areboth}

non-empty. If there isxE

X

such thatx inf

’? (respectively,

^{x sup}

:T),

^{thenone canshow}

by adirect argument that ^x sup

’ (respectively,

x inf

:T?).

^{Thusthestatementthat either}

x inf

.T?

orx sup

:T

issufficientto guarantee that

"

^{_0x.Therefore,if}

^(a)

^and

^(b)

^bothfail

to hold,then

’t

and

’l

mustbothbe non-empty and inf

:T

andsup

’l

mustbothfailto exist;

consequently,

(c)

^musthold.

Wedefineatotally orderedspace

(X, r)

to be^atotallyorderedset

X

equippedwithatopology rwhich islocallyconvexand

T2-ordered.

Foranytotallyordered set

X, (X, 0)

^{isatotallyordered}

space, and indeed0 is the coarsest topologyon

X

which is both

T2-ordered

and locally convex.

In

particular, ifris acompact,T2-ordered,locallyconvextopologyon

X,

thenitfollowsfrom the preceding statement thatr 0. Thus everycompact, totallyorderedspacehas the order topology.

Itshould be noted thatthe term"totally orderedspace" isdefined ina moregeneralwayhere thanin

[3],

wherethis term isapplied only to spaceswiththe order topology. Weshall normally designateatotally ordered space

(X, r)

^{simply by}

Foranytotally ordered space

X,

a singularity on

X

is defined tobe anynon-convergent, monotone,free, convexfilter, or, equivalentlyin viewof Proposition 1.2, theconvexhull ofany non-convergentultrafilter.

We

shalluseProposition^1.3todefineseveraldifferenttypes of singu- larities:

(1)

^{If jr} is a singularity such that jrT

(respectively,

^jr

),

^{then jr is an increasing}

(respectively, decreasing) unbounded, ^simple, singularity.

(2)

^{Ifjr is a}singularity such that jr _0_{xforsome xE}

X,

^{then jr} isa bounded, simple singu- larity.

(3)

^{Ifjr is a}singularitysuchthat jrT

#

^{and jr}

# ,

^{then jr} is an essentialsingularity.

Let

S (X)

^{betheset ofall}singularitieson atotally orderedspaceX. Wetotally order ^$

(X)

by imposing the relation: jr

.

^{iff there is}

^F

^{E jr such that}

^{F ft.}

^If

^X

^{has no greatest}

(respectively,

least)

element, then there is an increasing (respectively,

decreasing},

unbounded, simple singularitywhich isthe greatest

(respectively, least)

^elementin

.q (X).

^{Thus thereare at}

most twounbounded simplesingularities. Thenext proposition shows that the essential singu- larities alwaysoccur asorderedpairs.

Proposition

1.

^A singularity jr on a totally ordered space

X

is essential iffthere is asin- gularity

.

^{suchthat jr f3 is a}non-monotone,convexfilter. Ifjr is an increasing

{respectively, decreasing)

^essentialsingularity, then

.

^{is adecreasing}

(respectively, increasing}

essential singu- larityand jr

(respectively, jr).

Proof.

^{Letjrbe an}increasing, essentialsingularity. Recall that ^jrt andjrr aredecreasing and increasingsets, respectively, whichpartitionX. ^{Since jr}is essential, sup^jr and inf jrrboth fail toexist; thus jrt containsnogreatestelement andjrr contains noleastelement. Thus thefilter generated by

{(,--, a)

^{f3jrr a}

e jr}

iswell-defined, convex, andfree;

.q

islso

decreasin (since

jrr

gr g)

^andessential

(since

inf

gr

failsto

exist).

Furthermore^jrt jrand

(jr)r

^jrr E

g,

which implies^jr

< .

^{Finally,onecan}verify that ^jr

n

is anon-monotone,free,^convexfilter. If weassume,onthe otherhand,thatjris adecreasing,essentialsingularity,onecansimilarly show thatthefilter

.

generated by

{(a,-)

^fjranda

jr}

isanincreasing, essentialsingularity such that jr

n .

^{is a}non-monotone,free,convex filter and

. ^<

^jr.

Conversely, let ^jrbeasingularityandassumethatasingularity

.

^{existssuch that jr f}

.

^isa

non-monotone, convexfilter; ^jrf3

.

must also be freesince ^jr and

.

^areboth free. Thus ^jrf3

.

688 D.C. KENT AD T.A. RICIOND

hasthe formdescribed in Proposition1.1

(c),

^{and one}caneasilyshow that

and

rt .t ^{(y" n .)t.}

^If

^Y"

^{is notessential,then}

^Y"

z and this implies thatzE

(a,b)

for all aE

(y’.)t

and b

(Y’f)T. But

these

open"

intervals formabasefor

Y’,

^contrarytothe fact that

Y" n.G

^isafree filter. Thus

Y"

is an essentialsingularity, and the same reasoning appliesto

..

If

Y

^and

.

^areessentialsingularitiesonatotally orderedspace

X

such that

Y" .

^{isa non-}

monotone,convexfilterand if

Y" < .,

then the ordered pair

(Y’, .)

^willbe calledan essential pair singularities.

Let j0(X)

be the set of all such essential pairson

X.

Proposition 1.5

Let Y"

^bean increasingand

.

^adecreasing singularityon atotally ordered space

X.

Then

(/’, .) p(X)

iff/’

,r

^{and y’t}

g.

Proof.

If

(Y’, ) e(X),

^{then y’t}

.t

^and

^Y’t .t

^wasestablished in the proof of the precedingproposition. Conversely,

Y’r r

^{andy’t imply that}

^Y’r .r ^{(Y" n .)}

^and

Sets of theform

(a,--,) n y’t,

aG y’t formafilter base for

Y’,

andsets of the form

(,b) n .r,

^bE

.r

^{forma filterbase for}

^.;

^{unionsofsuchsets are}

"open"

intervalsof the form

(a, b),

^a

(Y" .)t

^{and b}

(Y’ .),

^andthese constituteafilterbasefor

Y" .

^Thus

Y" .

^is non-monotoneandconvex,and

(Y’, .)

G

p(X)

follows by Proposition1.4.

If

X

is the set ofrational numberswith the usual order and topology, then $

(X)

consists oftwo

unbounded,

simple singularities andanuncountable number of essential singularities; in this case there are no boundedsimple singularitiessince the usual topology is the order topoi- ogy.

Furthermore,

thereisanatural one-to-onecorrespondence between

p(X)

and the "irrational numbers." On the other hand, if

X

is the Sorgenfreyline

(i.e.,

^the real line with the usualor- der and half-openinterval"

topology),

then $

(z)

contains twounbounded simple singularities, uncountably many

bounded,

simple singularities,andnoessential singularities.

For

anarbitrary totallyorderedspace

X,

$

(X) @

^ifl"

^X

^iscompact.

Ordered Compactificationa.

If

Y

is aposet with atopology,

(Z, )

^{is a}topologicalcompactificationof

Y,

and

Z

^isalsoa poet,then

(Z, )

^isanordered compactification of the orderedtopologicalspace

Y

if theembed- ding

Y Z

is increasinginboth

directions

(i.e., z

< .v imvlies (z) < (.v)

andvice-versa.)

Weshallbeinterestedonlyin

T2-ordered

compactifications, which havethe additional requirement that

Z

be

T2-ordered.

Nachbin,

[4],

has characterizedthosespaces

(which

he calls completely regular ordered

spaces)

whichallow

T-ordered

compactifications.

In

particular, completelyregular orderedspaces must be

T-ordered

and locallyconvex.

A T-ordered

space is said tobe

T4-ordered (normally

orderedin

[4])

if, for eachpair

A, B

of disjoint closed sets such thatAisincreasing andBisdecreasing,there aredisjoint open setsUand

V,

withUincreasingandVdecreasing, suchthatA CUand

B

^CV.

Every T4-ordered, locally convex space is completely regular ordered

(see [2]).

Furthermore, it isasimplemattertoshowthat everytotally orderedspace is

T-ordered.

^Thuswehaveestablished Proposition ^$.1 Atotallyordered setXwithatopologyhasa

T2-ordered

compactificationiff

X

is atotally orderedspace.

In

general, ordered compactifications of totally ordered spaces need not be totally ordered unlessone imposes the restriction that thecompactificationbe

T2-ordered.

^{Recallthe following}

characterization of

T-ordered

spaces: If

Y"

x,

.

^y, and the productfilter

r

^{has atrace}

ontheorder,thenx

_<

y.

Proposition^$.$ A

T2-ordered

compactificationof anytotally orderedspace isatotally ordered space with the order topology.

Proof.

^Let

X

be a totally ordered space, and let

Y

be a compact,

T-ordered

^space

(not

necessarilytotally

ordered)

which contains

X

as adense subset. Ify,y2 E

Y X,

thenyl and

y2 are limits of singularities on

X,

and since

S (X)

^is totally ordered it follows that yl

_<

y2 or y

_<

yr. Ifx E Xandy C

Y-X,

then y is the limit of asingularity

Y"

^C

S(X),

^andsince

and partitionX x isin y’t _or

’.

Ifx

Y’r,

then

’

^{has atrace on}the order of

X (and

henceonthe orderof

Y).

^Since

Y

isT-ordered,y

_<

x. Similarly,if x

Y’l,

^thenx

<_

y. ThusY istotallyordered,andwerecall thateverycompact,totallyorderedspacehas the ordertopology.

The familiarprocedure for "ordering" the

T

compactificationsofa completelyregular

(non-

ordered)

space extends in a natural and obvious way to the

T2-ordered

compactifications of a

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

completely regular orderedspace. If

(Yl,a,)

^and

(Y2, or2)

^are

Tz-ordered

compactificationsof

X,

wesaythat

(Yl,a,)

()

(Yz,az)

^{ifthereis a}continuous, increasingfunction j’:

whichmakes thediagram

X Y

el

Y1

commute. Two

T-ordered

compactificationsof

X

are equivalentifeach is larger than the other in thissense.

Our goalisto describeall the

Tz-ordered

compactificationsofatotallyordered space, andwe beginwiththe largest,which iscalled theNachbin

(or Stone-Oech ordered)

compoctification

([2], [4]).

^{It turns out thatfor} totally ordered spaces, theNachbin compactification isequivalent to the Wallman ordered eompactification

[3],

^{and it willbeuseful} to give abrief description ofthe latter compactificationatthispoint.

Let Y be a

T-ordered

topologicalspace (partially ^butnot necessarily totally

ordered)

^with

a subbase of monotoneopen sets; it is shown in

[4]

that all completely regular orderedspaces

(and,

hence, alltotally ordered

spaces)

satisfy theserequirements. A subset

A

^of

Y

iscalled a c-set ifit isthe intersection ofaclosedincreasingset andacloseddecreasingset. Notethatevery c-set is closed and convex, but the converse is generally false. A

c-filter

^{is onewhich has filter}

baseofc-sets. Themaximalc-filterson

Y

form the underlyingsetfor

woY,

theWallman ordered compactification ofY.

ForanysubsetAof

Y,

let

A* {

6

woY A

6

’}.

The collection12*

(U*

^Uamonotone, open subset of

X}

constitutes an open subbasefor thecompactification topology of

woY.

The partialorderrelation on

woY

^{is definedby:}

" ^<_*

^iff

^{I( r)}

^C

.

^and

^D(.)

^_C

^’,

^where

^{I( r)}

^is

the filtergenerated by all closed, increasingsets in

r

^and

D(.)

^is generated by all the closed, decreasing sets in

..

The embedding

r ^Y woY

^{istheobviousone:}

r (y) ,

^forally

e

Y.

Proposition .8

Let X

beatotallyorderedspace.

The c-sets of

X

areprecisely the

closed,

convexsubsets.

(b)

^Thenon-convergent, maximalc-filters areprecisely thesingularitiesof

X,

and thus

woX {}:

^:

x} ^v s (x).

(c)

The Wallmanordered compactificationisthe largest

T:-ordered

compactification ofX.

Proof.

The first assertion is obvious. The second follows by first observing that eachsingu- larity hasafilterbaseofclosed,convexsets and then applyingProposition 1.2.

In

^{orderto prove}

(c),

it is necessaryandsufficient

(by

Corollaries 1.4and 1.5of

[3])

to show that

X

is

T4-ordered

andsatisfies the additional condition: Foreachclosed,convexsubset Aof

X, i(A)

^and

d(A)

^are

both closed. But ^theseconditionsclearly hold for totally orderedspaces.

Proposition

2.4

^Let

X

beatotally orderedspace. If

J’,

E

woX,

then

r

^<:,

.

^{iffexactlyone}

ofthefollowingistrue:

(a)

^Therearex,yE

^X

^suchthat

" , . ^y,

^{and x}

^_<

^{y in}

^X;

(b) .T

forsomex

X, .

^$

^(X),

^{and x}

^.;

(c) . ₌

^forsomey

^{X," e $(X),}

^{and y}

(d) J’,.

E

$(X)

^and

" ^< .

Lemrna 2.5Let

X

beatotally orderedspace. Then

(.T, .) (X)

^iff

, . ^e

^$

^(X)

^{and there}

isnoxE

X

such that

"

^{_<* _<*}

..

Proof.

Using Proposition 2.4,^{we see} that

,v <_, <_, .

^{is equivalent to z}

^rT

^N

..

^If

(r,.> (X),

^{then by} Proposition 1.5,

’T .T,

which implies

r?

f3

. ^.

^{Conversely if}

x

rt n ,

^then

^r? .

^isimpossible, andsoby Proposition1.5

(Y, .> ^p(X).

Proposition 2.6 Let

X

beatotally orderedspace. If

r, . ^woX,

^then

.

^covers

^r

^iffexactly

oneof the followingis true.

(a)

^Therearex,yE

X

such that

r , . ^y,

^{and ycoversx.}

(b) r

for some x

X, .

^{is a}decreasing, simple,boundedsingularity, and

.

^0_,x.

(c) .

^{for some x}

^{X, ,v}

^{is an}increasing, simple, bounded singularity, and

r

^0_,x.

(d) <r, ) (X).

692 D.C. KENT AND T.A. RICIOND

Proof.

^{Welimit the proofto thecasewhere}

.

^and

.

^arebothsingularities.

In

thiscase, ^it followsfromLemma 2.5 that

.

^covers

^.

^ill

^{(Y’, > e p(X).}

Since

woX

has the ordertopology foranytotally orderedspace

X,

wemayuse

Lemma

2.5and Proposition 2.6 to describethe basicneighborhoods in

woX

for each compactificationpoint. If

Y

^{is an}increasingsingularity

(either

simpleoressential, boundedor

unbounded),

^then "closed"

intervals in

woX

of the form

[, :T],

^{for x}

r,

forma base for the neighborhood filter at

:T.

Likewiseforadecreasing singularity

:T,

intervals in

woX

of the form

[r, ],

^{for x}

’T,

constitute a basicfamily of neighborhoods.

We next makeuseofourknowledgeof

woX

to describe anarbitrary

T2-ordered

compactifi- cationofa totallyordered spaceX. Since

(woX, x)

^is the largest

T2-ordered

compactification of

X,

any other

T2-ordered

compactification

(Y, )

^of

X

is related to the former by"

(Y, )(

(woX, ox).

^{Thusthere is an} increasing, continuous function

az woX - ^Y

^{which makes the}

diagram

commute.

X woX

We can think of

Y

as the quotient space of

woX

^obtained by identifying the compactification points

(i.e.,

^thesingularitiesof

X)

in an appropriate way.

Proposition 2.7 Let

(Y, )

^bea

T2-ordered

compactification ofatotally orderedspace

X,

and let

or "woX Y

^{be the}function described inthe preceding paragraph. If

r

^{and are}distinct singularities of

X

such that

r < .,

^then

a a;

^implies

(r, .) p(X).

Proof.

^If

<’,> p(X),

then by Lemma2.5there is x Xsuch that

"

^{_<* <_*}

..

^Since

ar

^{is an} increasing function and

az(r) ar(.),

^we must conclude that

ar(:T) ar().

^But

at(r)

^Y-

(X),

^whereas

ar() (X).

Thus the only possiblewaythat singularitiescanbe identified inordertoforma

T2-ordered

compactification

Y

as a quotient of

woX

^is to identify essential pairs ofsingularities to single

elementsin

Y. To

be moreexplicit, let

P

_C

p(X)

beasetofessential pairson

X,

and let Ypbe thetopologicalquotientspace obtained from

woX

^asfollows" for eachpair

<, .>

^E

^P,

^thedistinct

elements

Y"

and

.

ⁱⁿ

^woX

^weidentified toasingleelement,denoted by

(Y’, .>. ^Let

^ap

^woX

^Yp

bethecanonical quotient map;ap is continuousby construction, andweimpose onYptheunique total orderrelative towhichap isincreasing. Let

Cp X

Yp be thecompositionapo

x.

In

order to get a clearer picture of the

T2-ordered

compactification

(Yp, p),

we may iden- tifythe set Yp with

P’LJ P,

^where

P’ {/

E

woX

^{does not belong} to an essential pair in

P}.

^Yp consists of equivalence classes of members of

woX,

each ofwhich contains either one or twoelements. Thus

P’

consists ofthe elements of

woX

which determine singleton classes in Yp, whereas

P

can beidentified with thetwoelement classes inYp. If k’ E

P’,

^theupper boundsof

’

^{inYp consistof thoseelements in}

^P’

^whichare upper bounds of)4 in

woX,

alongwith those elements

(Y’,.) ^P

^such that

<_* ,z (or

equivalently,

_<* .)

in

woX.

^If

(Y’,.) P,

the upper bounds of

<,z,.>

^{in Yp consist} ofthose elements in

P’

which are above

r

ⁱⁿ

woX

^and

also those elements

(Y", .’)

^E

^P

^{such that}

Y"

<*

Y"

in

woX.

Forthoseelements ofYp in

P’,

the basic neighborhoods have thesameform in Yp as in

woX

^{except of}course, that the intervals involved mustbe construedas lyingin Yp insteadof

woX.

^{On the otherhand, for}

(Y’, .>

^E

^P,

the neighborhood filterinYp hasafilterbase of =closed intervalsinYpofthe form

[, .],

^where

<* :r

<*

.

If

P, Q

aresubsets of

p(X)

and

P

^C

Q,

then

Yo

^can be regarded as the quotient space of obtained by identifying thoseessential pairs

Y’, .

ⁱⁿ

^P’

^{such that}

^{(Y’, .)}

^{E Q. Thecanonical}

quotientmapapo Yp^--,

Y@

is continuousand increasing, and the quotient mapa₀

"woX Y@

is givenbya₀ ap_{0 o}crp. Ourresultson

T=-ordered

compactificatiormofatotally orderedspace

X

maybe summarizedinthefollowingtheorems.

Theorem .8 Let

X

be atotally ordered space, and let

P

be an arbitrary subset of

p(X).

Then

(Yp, Cp)

^{is a}

T2-ordered

compactification of

X.

If

P, Q

aresubsets of

(X),

^then

P C_ Q

itf

(Yo,o)((Y,,p).

If

P (,

then

(Yp,p)

^is equivalent to

(woX,ox)

and is the largest

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

T2-ordered

compactification of

X. If Q p(X),

then

(Y, CQ)

^is the smallest

Trordered

^corn-

pactificationof

X.

Theorem .9 If

(Y, )

^is any

Trordered

compactificationofa totallyorderedspace

X,

then thereis asubset

e

of

(x)

^suchthat

(I/’, )

^{isequivalentto}

CorollarF

^e.10

A

totally ordered space

X

has aunique ordered compactification iff

X

hasno essentialsingularities.

Examples of

Trordered

spaces withunique

T-ordered

compactification include the real line

(with

^{usualorderand}

topology),

^theSorgenfrey line, andthediscreteline

(the

real numberswith usualorderand discrete

topology). In

thecaseof theSorgenfreyline,one compactification point corresponding toeach real numberis

added,

alongwith leastelement -ooand greatest element oo. The

T2

ordered compactificationofthediscrete lineissimilar, except that two compactifica- tionpointsareadded for each real number

(one

oneach

side).

REFERENCES

I

BLATTER, J.,

"OrderCompactifications ofTotally OrderedToplogicalSpaces,"

J.

Approzi- marion TheorF, 13, 1975,58-65.

FLETCHER, P.

and

LINDGREN, W.F.,

Quasi-Uniform

Spaces, Lecture Notes

ⁱⁿ

Pure

and AppliedMathematics,

Vo.

77, Marcel

Dekker, Inc., New

York 1982.

KENT, D.C.,

"On the Wallman Order Compactification,"

Pacific

^Journal

of

Mathematics, 118, 1985, 159-163.

4

NACHBIN, L.,

Topology and

Order, Van

Nostrand Math. Studies,

No.

4, Princeton,

N.J.

1965.