On the Nachbin compactification of products of totally ordered spaces

VOL. 18 NO. 4 (1995) 665-676

ON THE NACHBIN

COMPACTIFICATION

OF

PRODUCTS

OF TOTALLY

ORDERED SPACES

D.C. KENT, DONGMEI LIU

Department

ofMathematics WashingtonState University

Pullman,

Washington, 99164-3113

and

T.A. RICHMOND

Department

of Mathematics

Western

KentuckyUniversity

Bowling

Green,

Kentucky42101

(Received October 18, 1993)

ABSTRACT.

Necessary

and sufficient conditions aregiven for

o(X

Y)

oX

x

oY,

where

X

and

Y

aretotallyordered spacesand

oX

denotes the Nachbin

(or Stone-ech ordered)

com-pactificatio of

X.

KEY

WORDS

AND

PHRASES. Nachbincompactification, Wallmanordered compactification,

totallyordered space, maximalc-filter,strictly firstcountablespace.

1980AMS

SUBJECT CLASSIFICATION

CODES:54

D

35, 54

F

05,54

B

10.

1.

INTRODUCTION.

If

X

is a

T3.5-ordered

space

(i.e.,

anordered topologicalspace which is %ompletely regular ordered" inthesenseofNachbin

[8]},

then

X

hasalargest

T2-ordered

compactification

oX

which iscalled the Nachbin

(or Stone-ech ordered)

compactification. Thiscompactification, introduced

in

[8],

hasbeen investigatedinallofourreferences

except

[5].

We

are interested in determining when

o(X

x

Y}

oX

oY,

a problem which has not previously received attention in the literature. The methods used by Glicksberg

[5]

to solve the corresponding problem for the

Stone-(ech

compactification do notappeartobefruitful when applied to theNachbincompactification.

Therefore,

atthispreliminary stage ofourinvestigation,

wehavefocusedour attentionon the casewhere

X

and

Y

are "totally ordered

spaces",

where

atotally orderedspace is definedtobe atotally ordered setwith aconvex,

T2-ordered

topology

Our solution to the aforementioned problem makes extensive useof the Wallman ordered compactification

woX,

introduced in

[2].

In

Section3weshow thatforany

T3.-ordered

space

X,

oX

canbe obtained from

woX

via acertain quotientconstruction, andthis

result

is employed

inthe proofofour maintheorem.

We

also makeuseof the fact that

woX

[oX

foranytotally orderedspace

X.

We

firstproveapreliminary version ofourproducttheorem in Section 4under the assumption

that thetotally orderedspaces

X

and

Y

are %trictlyfirstcountable".

(A

totally orderedspace is definedtobe strictly

first

countableifeveryneighborhoodfilterandeverymaximal closed-convex

filterhasacountablefilter

base.)

Surprisingly,the condition which works" in the strictly first

countablecase also works" in the general case

(see

Theorems 4.4 and

5.6).

If

X

and

Y

are

strictlyfirst

countable,

then

o(X

x

Y)

[oX

[oY

and

Wo(X

Y)

woX

woY

areequivalent statements.

We

do not know ifthisequivalence holds for arbitrary totally orderedspaces

X

and

Y.

2.

PRELIMINARIES.

Let

{X,_<)

be aposet.

Fora

non-empty subset

A

of

X,

we define

d(A)

{y

e

X

y

<

xforsomex E

A}

to be the decreasing hull of

A;

the increasing hull

i(A)

is defined dually.

We

shallwrite

d(x}

and

i(x}

in place of

d({x}}

and

i({x}}.

A

set

A

is i,creasing

(respectively,

decreasing)

if

A

i(A}

(respectively,

A

d(A}};

asetwhich is either increasingordecreasing is said tobe monotone.

For

any

A

C_

X,

A

^

i(A}

N

d(A}

iscalled the convexhull of

A,

and

A

is convexif

A

A^.

Let

F(X)

denote the set of allfilters on aset

X. We

alwaysuse the term

filter

o

mean a

proper set filter.

If

.T

and

.

arefilters on

X

such that

F n

G

#

,

for all

F

E

Y

and

G

C

.,

then

r

y _{denotes thefiltergenerated by}

(F

NG"

F

e ’,

G

e .);

if,ontheother

hand,

containdisjoint sets,wesay that

"

v

.

[ails

to exist.

A

filter

"

is

]tee

ifthereis nopointcommon

to allthe sets in

.T.

A

filter which is notfreeis said tobe fixed; in particular, thesymbol will

denote thefixed ultrafiltergeneratedbyx G

X.

For

any filter

Y

on

X,

the filter

’^

generated

by

setsofthe form

{F

^

F

E

.T}

iscalled the convexhullof

’.

An

orderedterpalogicalspace,orsimplyan orderedspace, isatriple

(X, _<, r),

where

(X, <)

is a

poser

andr a eanvextopologyon

(X, <_}

(i.e.,

r is atopology which

has

a subbase consisting

ofmonotone open

ets).

Note

that every ordered spaceis locally convexin the sense thatevery neighborhood filterhasabase ofconvexopen sets. Whenthereis nodanger of confusion,werefer tothe ordered space

(X,

<, r}

simply as

X.

If

X

and

Y

areorderedspaces,amap

f

X

Y

is

increasing

(respectively, decreasing)

if x

<

y in

X

implies

f(x) <_

f(y) (respectively,

in

Y.

A

continuous,increasing map iscalled anordered topological morphism, ormorebrieflya

morphisrn.

A

b]jective morphism whoseinverse isalsoamorphismiscalled anordered $opological

isomorphism, or morebrieflyanisomorphism.

Let

VI’(X)

(respectively,

CD’(X))

denote theset

of all morphisms

(respectively,

continuous,decreasing

maps}

fromanorderedspace

X

into

[0, 1].

An

ordered space

X

is

T-ordercd

if

i(x}

and

d(x}

areclosed sets, forall x

X;

X

is

T-ardered

if the partial orderrelation

_<

isclosedin

X

x

X. An

orderedspace

X

is

T.s-ordered

(completely

regular orderedin

[8])

if thefollowing conditionsare satisfied:

(1}

Ifx

X, A

is aclosed subset of

X,

and x

A,

then there is

f

CI’(X)

and g

CD’(X)

such that

f(x)

g(x}

0and

f(y) v

g(y)

1, for all y E

A;

(2)

Ifx yin

X,

there is

f

CI’(X)

such that

/(y}

0 and

f(x)

1. The

T3.-ordered

spaces are precisely thosewhich allow

T-ordered

compactifications

(see

[3]

and

[8]}.

An

ordered space

X

is normally ordered

(see

[8]}

if, whenever

A

and

B

are

with b increasing andV decreasing, such that

A

C_ U and

B

C_ V.

An

ordered space which is

both normallyordered and

Tl-ordered

issaidto be

T4-ordered.

Given a

Ts.s-ordered

space

X,

thereis alargest

T2-ordered

compactification of

X

called the Naehbin

(or Stone-ech ordered)

cornpaetification,denotedby

[3oX.

Thestandardconstructionof

/9oX

involvesembedding

X

in the "ordered cube"

[0,

1]

ct’(x)_{where the latter space has theusual}

productorder and topology. Thiscompactificationischaracterizedbythe followingwell-known

exnion

orem

Oee

[]

or

[]).

THEOREM

2.1. If

X

isa

Ts.s-ordered

space,

Y

isacompact,

T2-ordered

space,and

f

X

Y

isamorphism, then thereis aunique morphism j-r

./oX

y

such that the diagram

X

ex

\ Y

commutes,

where

ex X --/3oX

is thecanonicalembedding.

We

next review the construction ofthe Wallman ordered compactification.

Let

X

be an

oderdp=e

ana

X;

e

()

(repey,

())

b

h ml dod,

dr=.g

(re-spectively,

closed,

increasing)

set that containsA.

Let

A

a

I(A)

q

D(A);

if

A

A

a then

A

is calleda e-set.

One

mayverify that thecollection of alle-sets on

X

is closed under arbitrary

intersections.

It

is obviousthat e-sets are dosed and convex, but not all

closed,

convexsubsets of

X

aree-sets. IfjrG

F(X),

let

D(jr) (respectively, I(jr))

denote thefilter on

X

generatedby

{D(F)

F

G

jr} (respectively,

{I(F)

F

jr}).

The filter jra

1(jr) v D(jr)

isgenerated bysets

of the form

ava,

for av jr; ifjr

jra,

thenjr iscalled a

e-filter.

One may easily verify

(using

Zorn’s

Lernma)

that every e-filter iscoarserthan amaximal e-filter.

Let

X

bea

Trordered

space, andlet

WoX

be thesetof allmaximal e-filterson

X. A

partial

order

"<"

for

WoX

is definedasfollows: jr

<

=

I(jr)

C

#

and

D()

jr.

We

also assign to

woX

the topologywithclosed subbase

g

{A

A

aa},

where

A

Then

woX,

withthe order and topologyjustdescribed,is anorderedspace which iscompactand

T

(but

generallynot

T-ordered).

If

x

X

WoX

isdefinedby

x(z)

5,

for all zG

X,

then

x

is anisomorphic embedding,and consequently

(WoX, x)

is anorderedeompactificationof

X. Furthermore,

wehavethefollowingextensionproperty

(see

[2]

or

[6]).

THEOREM

2.2.

Let X

bea

T-ordered

space,

Y

acompact,

T2-ordered

space,and

f

X

--

Y

amorphism. Thenthereisaunique morphism

jz. w,X

Y

such that the diagram

X

Ox

woX

\

Y

commutes.

An

orderedspace

X

isdefined tobeac-spaceif

i(A)

and

d(A)

areclosed subsets of

X

when-ever

A

_C

X

is ac-set. The next theoremisprovedin

[6].

(I)

woX

is

T-ordered

()

,,,ox

oX

(3)

X

isa

T4-ordered

c-space. 3.

/oX

AS

A QUOTIENT OF

woX.

Throughout this section,

X

will denote an arbitrary

Ts.s-ordered

space and

(woX,)

the

Wallmanorderedcompactificationof

X.

If

f

E

CI’(X)

then there exists, byTheorem 2.2, a unique morphism

f^

woX

[0,

1]

extending

f.

We

define an equivalence relation on

woX

as follows:

{(r, )

E

woX

x

woX

f^(’)

f^(.),

forall

f

CI’(X)}.

The set

{[}’]

"

6

woX}

of E-equivalence

classes is denotedby

woX/;

let a

woX

woX/

be the canonical projection map.

We

en-dow

woX/

with the quotient topology derived from

woX

and a, and with the partial order

[Y’]

5

[.]

=>

f^()

<_

f^(.)

in

[0,1],

for all

f

6

CI’(X).

Theset

woX/,

with this order andtopology,is anordered spacewhich,forconvenience,weshallcall

#oX.

One easily verifies thata

"woX

oX

is amorphism.

LEMMA

3.1.

oX

isacompact,

T-ordered

space.

PROOF.

Obviously,

#oX

iscompact.

We

recall

(see

[8])

thatanorderedspace is

T-ordered

wheneverz y, there are disjoint neighborhoods

U

and

V

ofxand y, respectively, such that

U

isincreasing and

V

isdecreasing.

For

each

f

CI’(X),

define

f’:

oX

[0, 1]

by

f’([])

fA(),

for all

[ff]

6

oX;

it is eyto verify that

f’

is awelldefinedmorphism, nd therefore

(f’:

f

UI*(X))

CI*(oX).

[]

[],

there is

f

UI’(X)

such that

f()

f()

in

[0, 1],

d hence

f’([Y])

>

f’([])

in

[0, 1].

Let

f’([])- f’([])

>

0, d let

U

(ff)-((ff([])- e/3,1])

and

V

(f’)-([0, ff([])

+

e/3)).

Then

U

d

V

edisjointopennieghborhoods separating

[Y]

and

[]

such that

U

is increing d

V

decreeing. Therefore

poX

is

Tz-ordered.

By

Theorem 1.2, thereis aunique morphism

’woX

oX

suchthat the diagram

/

x

\

oX

coutes.

Also,

for y

f

CI’(X),

thereisaunique

f

CI’(oX)

such that

f

f

oe.

Note

that

fA

f

o

,

sincetheemaps agreeonthe dense subspace

(X)

of

woX,

d henceon

woX.

LEMMA

3.2. For

Y’,

.

woX,

(Y’,

)

.(Y’)

(.).

PROOF.

If

(.,,)

,

then

f^(r)

f^(Q),

for all

f

C!(X),

and hence

f(())

/(())

for all

f

CI’(X).

This implies

eCY)

(),

since

oX

is

T3.s-ordered

and hence

eI’(ZoX)

{f

f

uP(X)}

separates points in

oX.

Conversely, if

()

(),

then

fA()

f(@()) =/(())

fA()

for all

f

CI’(X),

which implies

(, )

.

PROOF. Let

e’’#oX

floX

be definedby

e’([’])

g’(’),

for all

woX.

woX

-&

#oX

x

/

%

I"

%

ox

I

follows from Lemma 3.2 tha

"

is a well-defined bijection. Since

#oX

h the quotient

topology inducedby and

"

o iscontinuous, e" iscontinuous. Since

#oX

is

compac

and

oX

is

T,

"

is ahomeomorphism.

To

cheekthate’isanorder-isomorphism, let

[Y]

[]

in

#oX.

Then,

for all

I

C

CI’(X), ()

I()

i

[0, 1,

a

h

[(())

((5)),

fo

I

cI’(X).

Th

()

()

i

oX,

andconsequently

e’Ca(7))

e’([T])

e’([])

e’Ca()).

This argument is reversible, and

consequently both e" d

(e’)

-

are increing maps.

4.

A PRELIMINARY PRODUCT

THEOREM.

Compactificationsoftotallyordered spacesarestudied in

[I]

and

[7],

andwebegin this section

bysummarizingsomerelevent results from

[7].

We

definea

totall!/ordered

space

X

to be a

T2-orderedspace whose partial order is atotal order

(i.e.,

ifz,/E

X,

then z

<_

/or l/

_<

z).

It

is

easy toshowthatatotally orderedspace isa

T4-ordered

c-space in which the c-sets areprecisely the

closed,

convexsets. Consequently, by Theorem 2.3, the compactifications

woX

and

floX

ofa

totally orderedspace

X

exist andareequal.

Furthermore,

every

T2-ordered

compactificationofa

totally orderedspace is itselfa totally orderedspace.

For

atotally orderedspace

X,

we usethe equivalence of

floX

and

woX

to describethe com-pactification pointsof

[3oX

asmaximal c-filters.

It

isshownin

[7]

that,

inatotally orderedspace

X,

themaximal c-filters arepreciselytheconvexhulls of ultrafilters; the non-convergentmaximal c-filterson

X

arecalled singularities. Givenasingularity

,

let

T

t3{FT

F

E

7},

where

F

denotes the set ofupperbounds of

F,

and

yt

U{Ft

F

C

},

where

F

is the set oflower boundsof

’.

Theconvexsets and)’t partition

X,

andsoexactlyoneof these sets is in

7.

If

’t

"

(respectively,

’t

’)

wesay that

.

isa decreasing

(respectively, increasing)

singularit!t.

A

totally ordered space

X

is

strictll

first

countable if every neighborhood filter and every

singularity has acountablefilter base. If

.

is an increasingsingularitywith a countablefilter

base,

thenthere is astrictly increasingsequence Zl

<

z

<

z3

<

in

X

such that is the

convexhull of thefilterofsectionsof

(z);

similarly, each decreasing singularitywithacountable

filter base is likewise derivedfromastrictlydecreasingsequence in

X.

If

X

and

Y

are totally ordered spaces, then

X

x

Y

(with

the product order and product

topology)

is a

T3.s-ordered

space, but not generally a c-space. For instance, it is shown in

[4]

thatif

X

is thereal line withthe usual order and topology and

Y

is anytotally orderedspace

whose underlying

poser

isthereal line, then

X

Y

isa c-space the topology for

Y

isthe usualtopology.

Thus,

in general,

wo(X

Y)

fails tobe

Tz-ordered

and hence

wo(X

x

Y)

and

[3o(X

Y)

arenon-equivMent eompaetifieations

(see

Theorem

2.3).

The next twolemmasaredue to

Margaret

A. Gamon.

LEMMA

4.1. If

X

and

Y

aretotallyordered spaces,

A

ac-set in

X,

and

B

ac-setin

Y,

then

A

B

isac-set in

X

Y.

i(A

B)

I(a)

I(B)

is a

closed,

increasing set containing

A

x

B,

and hence

i(a

B)

I(AB)

I(A)I(B).

Similarly,

D(AxB)

D(A)D(B),

and therefore

I(AxB)nD(AB)

(I(A)

x

I(B))63 (D(A)

D(B))

(I(A)n D(A))

(I(B) D(B))

A

B.

Thus

A

x

B

is a c-set in

X

x

Y.

LEMMA

4.2.

Let

X

and

Y

be totally orderedspaces, and let jr be a singularityon

X

and

.

a singularityon

Y

suchthat either both are increasingsingularities or both are decreasing

singularities. Thenjr

.

isamaximalc-filteron

X

Y.

PROOF.

Assume that jr and are both increasing singularities on

X

and

Y,

respectively.

By Lemma4.1,

jr

6

is a c-filter on

X

Y.

Let X

{x

E

X

<

jrinwoX}

and

Y’

{y

E

Y

<

in

ToY}

be totally ordered subspaces of

X

and

Y,

respectively.

Let

jr’ and

betherestrictionsofjr and

.

to

X

and

Y,

respectively.

It

is easy to verify that jr and

.

are increasingsingularitieson

X’

and

Y’,

respectively,and that

(jr,}r

(06’}r

0.

It

is alsoeasy to verify that jr

.

is amaximal c-filteron

X

Y

==

jrw

6

is amaximal c-filteron

X

Y.

Therefore weshall assume, without loss of generality, that jr and are increasing singularities on

X

and

Y,

respectively, such that jrr

0.

Let

be a c-filter on

X

Y

such that jr

.

C_ )/.

Let H

)/ be a convex set, and let

(a, b) H63(F G),where

F

jr_andG

.

For

anyeEXandd Ysuchthat

a<c

and

b

<

d,wemusthave

(c, d)

C

H,

sinceotherwise

i(c)

i(d)

would beamember ofjr

.

disjoint fromH. This implies that

i(a)

i(b)

C_ g.

But

i(a)

i(b)

Cjr

,

andso C_ jr

.

It

follows that jr

.,

andconsequently jr is amaximalc-filteron

X

Y.

In

general,the conclusionof

Lemma

4.2isnotvalid ifoneof the singularities is increasingand the other decreasing.

Indeed,

the next lemmaestablishes that ifjr is anincreasingsingularity

on

X

and

,

adecreasing singularityon

Y,

bothwithcountablefilter

bases,

thenjr

x

.

isnota maximal c-filteron

X

x Y.

LEMMA

4.3.

Let

X

and

Y

be totally orderedspaces.

Let

(x.)

be astrictly increasing se-quence in

X

and

(y,,)

astrictlydecreasingsequence in

Y.

Let S

{(x2r,-l,

y2,,_)’rt e

N)

and

T

{(xz,

yz,.,)’n

G

N}.

Then thereisg

e

CI’(X

Y)

such that

g(i(S))

1 and

g(d(T))

O.

PROOF.

Choose

xo

X

suchthatXo

<

x.

(In

case

x

istheleast element of

X,

replace

the original sequence

(x)

by

(x’),

where

x

x.+,.)

Since

X

and

Y

areboth

T4-ordered

spaces,we canapply Theorem 1, page30,

[8]

to obtain, for each rt C

N,

afunction

fzr,-

CI’(X)

such that

fz,.,-(a)

0 ifa

<

x2._ and

f,.,_(b)

1 if b

>_

x._, and another function

f,.,

CI’(Y)

such that

f2,.,(c)

0ifc

_<

yzand

f(d)

1 ifd

We

nextdefineafamilyoffunctions in

CI’(X

x

Y)

asfollows"

H(,

)

(fxC)f())

^

H4Cx,

y)

[g(x,

y)

+

f3(x)f4(y)]

A 1

H,,.,(x,

y)

For

k C

N,

wedefineS2k-t

{(xn-t,

Y2n-1)

_n

<

k)

_and

T

2t:

((x._,.,,

y2.)

_n

<

Finally, let

(,

)

[((,

)

+

f+C)f+())

].

It

isclear thatg is an increasing mapfrom

X

Y

into

[0,

1]

suchthat

g(z, y)

1 if

(z,

y)

E

i(S)

and

(z,

y)

0if

(z,

y)

E

d(T).

By

consideringthe possiblecases,onemayalso verify that forevery

(z,

y)

X

Y,

#(z,

y)

is theinfimumofthe constant function 1 and afinitesumof

continuousfunctions. Thusg

CI’(X

Y),

asdesired.

In

the proof of the next theorem we willneed some additional notation.

Let

X

and

Y

be totally orderedspaces, andconsiderthe following diagram:

e

oCXxr)

#/z

woCXxY)

XxY /

a’

XxY

-a

oXxoY

’,

woXxwoY

where,

inthenotationof Section2,e exxY, a

ex

x er, #xxr,and #xx#r arethe

canonicalembeddingmaps. Since

woX

oX

and

woY

oY,

oX

x

floY

woX

x

woY

is a

compact,

T2-ordered

space, and

a’

and 9 arethe unique,

continuous,

increasing extension maps

whoseexistence is guaranteedin Theorems2.1 and2.2.

Observe that

/(X

x

Y)

/oX

x/og

(respectively,

wo(X

x

Y)

woX

x

wog)

o’

(respectively,

)

isaninjective map.

THEOREM 4.4. If

X

and

Y

are strictly first

countable,

totally ordered spaces, then the

following statementsareequivalent.

1)

woX

x

woY

wo(X

x

Y).

(2)

oX

x

oY

:/o(X

x

Y).

(3)

Ifeither

X

or

Y

hasanincreasing

(or

decreasing)

singularity, then the other space con-tainsnostrictlydecreasing

(or

strictly

increasing)

sequence.

PROOF.

(1)

=

(2).

For

totallyordered spaces

X

and

Y,

IoX

woX

and

oY

woY.

Thus

wo(X

Y}

oX

oY,

and the latter space is

T2-ordered.

By

Theorem2.3,

wo(X

Y)

o(X

x

Y)

oX

x

Y.

(2)

=

(3).

Assume X

hasanincreasingsingularity

.q

and that

Y

containsastrictly decreasing

sequence

(yn).

Note

that is acompactificationpoint in

oX

woX.

If

(y,)

converges to yo in

Y,

then we_{define "7}

(,or(yo));

if

(y,)

fails to converge in

Y,

let

N

be the decreasing

singularityin

Y

which isthe convexhull of thefilterofsectionsof

(,),

and let7

(.q,)/).

In

eithercase,"7 is acompactificationpointof

oX

x

oY.

Next,

let

(z,)

be a strictly increasing sequence in

X

obtained

by

choosinga

denumerable,

nested filter base

{G,

:nE

N}

for

.

and choosing z, E

,,

for all n6

N,

such ha

z

<

z

<

za

<

Then clely isheconvhull of thefilerofsectionsof thesequence

(z,).

Csiderhefiler ofaecfionsof he sequence

(z,,

y,)

on

X

x

Y;

le behe filer generated by he subsequence

(z,-t,

y-)

d he filergenerated

by

he subsequence

(,,

y,).

If S

{(2n-,, Y2n-1)

:

N}

d

T

{(x,,

y,):n

N},

then S and

T

6

.

Regdless of whetherornot

(y,)

converges in

Y,

a(Y) (d

hence also

a(fa))

convergesto in

oX

x

oY.

Let

1

d bemimal c-filterson

X

x

Y

such that

a

and

a

.

By

Lena

4.3, thereis g

e

CI*(X

x

Y)

such that

g(i(S))

1 d

g(d(T))

0. Since

g-l({1}) e 1

and

-equiva|encec|asses

[i]

and

[.12]

(defined

inthe second paragraphof Section

3)

as elements of

fl0(X

x

Y).

Since

g^(l) # gA(2)

(where

gA

CI’(wo(X

x

Y))

is the unique extension of g

CI’(X

x

Y)),

[]

and

[]

are distinct equivalencecloses

(i.e.,

distinct elements in

o(X

x

Y)).

But

a(,)

and

a()

are both finer than

a(a),

soboth ofthesefilters converge to in

fl0(X

x

Y).

Thus

a’([])

a’([]),

andconsequently

a’

isnot injective.

(3)

(1).

Ifneither

X

nor

Y

has a singularity, then

X

and

Y

are both compact, and so

X

x

Y

woX

x

ToY

wo(X

x

Y).

Assume that

woX

x

ToY

contains a compactification point

(,).

We

will show that

-(q)

is asingletonin

wo(X

x

Y),

implying that is injective. Therearethree possibleces to consider. If is a singularity in

X

and a singularity in

Y,

then Condition

(3)

and the sumptionofstrict firstcountabilityfor

X

and

Y

imply that and are either both incre-ing sincre-ingularitiesorbothdecreeing singularities.

By

Lemma

4.2, x is amimal c-filteron

X

x Y.

Thus

-(q)

{7

x

}

is a single compactification point in

wo(X

x

Y).

If

is a

singularity and

r(Y)

forsomey

Y,

thenoneeilyverifiesthat

7

x

9

is anon-convergent

mimal c-filteron

X

x

Y,

and

-()

{

x

9}

is again a singleton compactificationpoint in

wo(X

x

Y).

Thesamereoning appliesif isasingularityin

Y

and

x(x)

forsomex

X.

We

conclude that the quotient map

-’wo(X

Y)

woX

x

ToY

is injective, and therefore

wo(X

x

Y)

woX

x

woY.

If

X

isthereal line with anyconvex,

T2-ordered

topology and

Y

is anystrictlyfirst

countable,

totally orderedspace, it follows from thepreceding theoremthat

/o(X

x

Y)

oX

x/oY

iff

Y

is finite. If

N

is the set ofnatural numberswith the usualorder and the discrete topology,

o(N

x

N)

oN

x/oN

follows by Theorem 3.4; notethat

N

x

N

is notpseudo-compactand

thus/(N

N)

N

x

/N. If,

on the other

hand, N

is the set of negative integers with theirusualorderand thediscrete

topology,/o(N

x

N

)

/oN

x/oN

.

Indeed,

it is easy tosee

that/oN

x

oN

hascardinalitybo,whereasit canbeshown

that/o(N

x

N

has cardinality 22

5.

THE

PRODUCT THEOREM.

Throughoutthissection,thesymbols

X

and

Y

willrepresentarbitrary totally orderedspaces.

Ourmaintheorem

(Theorem 5.6)

establishesthat Condition

(3)

of Theorem4.4

(stated

inslightly different

terms)

is necessary and sufficient

for/o(X

x

Y)

oX

x/oY

in thegeneralcase. The

proof ofTheorem5.6 isbasedonfiverather technical

lemmas,

forwhichweneedsomeadditional notationandterminology.

Let

"

be an increasing singularityon

X

and

f

anordinal number. Recall that

"

is alsoa

compactificationpoint in

woX

IoX.

We

saythat

,z

has order ifthereis astrictly increasing net

(x^)^<

on

X

suchthat the net

(^)^<

in/oX

converges to

r,

and istheleast such ordinal. The order ofadecreasing singularityon

X

isdefined dually. If is the order of any singularity on

X,

then clearly isaninfinite ordinal andtheleast ordinalofitscardinality.

In

astrictly first

countable,

totally orderedspace,everysingularity has orderw, the leastinfinite ordinal.

Next,

lety E

Y.

Ifnostrictlyincreasingnetin

Y

convergesto y, we saythat yhas

left

order

0. Ifforsomeordinal

,

there is a strictly increasing net

(y^)^<

converging to y in

Y

and is

the least such ordinal,wesaythatyhas

left

order

.

The right order fory is defineddually. Ifz E

Y

and y

<

z, let

[y,z]

{a

Y

"y

<

a

<_ z}, [y,z)

{a

Y

y

<_

a

<

z},

and

net converging to y, we denoteby

"l)t(y)

the filteron

r

generated by

{[y^,y]

A

< p};

"lit(y)

is

called the

left

neighborhood

filter

aty,andwe set

"t(y)

in case p 0. Likewise,if yhas right

order

>

0 and

(z^)^<

is astrictly decreasingnetconverging to y, the right neighborhood

filter

V.(y)

is generated by

{[y,z^]

A

<

};

againweset

Vr(y)

if 0.

Furthermore,

if

>

0 wedenote by

);(y)

the filter on

Y

generatedby

{(y,z^]

A

<

}.

Note

that

)t(y) n Yr(y)

isthe usual neighborhood filter at y.

We

shall also needadditional interval notation pertaining to singularities.

If

jrisanincreasing singularityon

X

and zE

X

is such that $

<

jr in

woX,

we define

[z,

jr)

{a

E

X

z

<

a in

X

andh

<

jr in

woX}

and

(z,

jr)

[z,

jr)\{z}. In

ease is a decreasing singularityon

Y

and

y

Y

is suchthat

.

<

in

woY,

let

(.,y]

{a

Y"

a

<

y in

Y

and

.

<

h in

woY}

and let

(.q,y)

(.,y]\{y}.

Ifjr has order and

(z^)^<

is astrictly increasing net suchthat

(:^)^<

converges to jr in

woX,

theneachof the sets

{Ix^,

jr)

A

<

}

and

{(z^,

jr)

A

<

(}

is a filter basefor jr. Likewise, if

.

has order rt and

(y^)^<.

is a strictly decreasing net in

Y

such that

(0h)^<.

converges to

.

in

woY,

then each ofthesets

{(.,U^]

A

<

r/}

and

{(.,U^)

A

<

r/}

are filterbases for

.

LEMMA

5.1.

Let

jrbeanincreasingsingularityon

X

oforder

>

w,let

.

beadecreasing sin-gularityon

Y

of orderr/> w,and assumethat every strictly increasing sequenceon

X

and every strictlydecreasingsequenceon

Y

isconvergent. If/ and aremaximal c-filterson

X

Y,

both finerthanjrx

6,

then for all

f

CI(X

x

Y),

f()

and

f()

convergetothesamelimit in

[0, 1].

PROOF. Suppose

thereis

f

CI’(X

x

Y)

suchthat

f(}

converges toa,

f()

converges to

b,

anda b in

[0,

1].

Let U

and

V

be disjoint neighborhoodsofaand

b,

respectively,and choose closed sets

L

E and

M

such that

f(L)

_C

U

and

f(M)

C_

V. Then,

choosethefollowing

points in

X

Y

Continuinginthis wayweobtainsequences

(a.,

b,.,)

in

L

and

(c., d.)

in

M

suchthat

ao <

co <

al

<

Cl

<

< an < cn

<

bo

>

do

>

bl > d >

>

b,

>

d,,

>

Under the assumptions of the

lemma,

the sequence ao,co,al,cl, converges to some Xo in

X,

and the sequence

bo,do,

b,d,..,

converges to some yo in

Y.

Thus

(xo,

yo)

_

L

f3

M,

contrary to the fact that

f(L)

f(M)

.

LEMMA

5.2.

Let

jrbeanincreasing singularity of order

>

won

X,

andlet y

Y

haveright

orderr/>w.

Assume

thateverystrictlyincreasing sequenceon

X

andeverystrictly decreasing

sequenceon

Y

isconvergent.If and aremaximalc-filterson

X

Y,

bothfinerthanjrx

)(y),

then forall

f

CI(X

Y),

f(.)

and

f(t)

converge tothesamelimit in

[0, 1].

LEMMA

5.3. Let beanincreasing singularityon

X

of order

>

wand let y

Y

haveleft

order rt :> 0. If isamaximal c-filteron

X

Y

finer than

27

e(y),

then

.M

.T

.

PROOF.

Let

(Yu),<

beastrictly increasing neton

Y

converging toy; thus

)t(Y)

hasafilter

base

{[Yu,Y]

#

<

r/}.

Note

that

27

hasafilter baseof the form

{Ix^,

27)

A

<

},

where

(x^)^<

is astrictlyincreasingnetin

X.

Choose ac-set

M

.M

suchthat

[Xo,.7")

[yo,y]

_

M. For

eachA

<

and# <

r, there is

(a^u,b^,)

M

N

(Ix^,

:7)

[y,, y]).

Let

(a, b)

M

be arbitrary;weshall show that

(a,

y)

M.

Ifb y there isnothingmoreto show,so assume the contrary,and chooseordinalsp

<

r/ and

r

<

such that b

<

yp

anda

<

x,. Then

(a, b) < (a,

b,p)

< (a,a, b,a),

andsobyconvexityof

M,

(a,b,,)

M.

Since

y,

<_ b,.,, (a,y,,)

M.

Indeed this reasoning implies that

(a, y,)

M

for

all

<

rt suchthat p

<

u.

Since

M

is

closed,

(a,

y)

M.

Using again the convexityof

M,

we

deducethat

[a,

27)

x

{y}

C_

M

and that

[a,

27)

,v.

Thus

.7

>

,

andsincebothare maximal

c-filters,equality holds.

LEMMA

5.4.

Let

"

be an increasing singularityon

X

of order

_>

w, and let y

Y

have right order _rt

>

w. If _{rt and} is amaximalc-filteronX

Y

finerthan

’x

r(y),

then

,=

x

.

PROOF. Let

(x)<e

bea strictly increasing net in

X

such that

{[x,.v)

A <

}

is a filter base for

’.

Let

(Yv),<,

beastrictly decreasing netin

Y

converging toy such that

{[y,y,]

<

isafilterbasefor

"Vr(y).

Let M

)[ bea

closed,

convexsetsuch that

M

C_

[Xo, .7)

[y,

yo].

CASE 1. rt

<

.

If0

<_

A <

rt, choose

(ax,b,)

M

N

([xA,.T)

[y, yA])

such that isstrictly increasing in

X

and

(bx)A<

is strictly decreasingin Y.

Next,

choose ordinal p such that

r

<

p

<

,

and choose

(aa,ba)

M

N

[xa,

)

[y, yo]

such that

a

<

ap, for all

A

<

r/. Let

A

{A

<

rt

"bx <

ba}.

Using the convexity of

M,

(a,,b,)

M,

for all

A

A

and

(a,,b,)

M

implies that

(a,,,bx)

M,

for all

A

A. Since

(bx)eA

converges to y in

Y

and

M

is

closed,

(ap, y)

M. This reasoning leads to the conclusion that

Ida, )

{y}

C_

M,

and hence

27

>:

CASE 2.

<

ft. Foreach

A

<

r,choose

(a,,b,)

M

[Xo,’)

x

[y,y,]

such that

(bx)A<,

is strictly decreasing; thus

(bx)<

nconvergestoy inY. Foreach

A <

rt,let

ux

bethe least ordinal

such that

a

<

x.

Note

that

{

A

<

rt}

C_

{p

p

<

}.

Considering thenet

(x,)<,

we observe thatsince each#

<

,

there is some

{A

A <

#}

suchthatthe term

xa

occurs

times in the net

(x,)<,,

where

Irtl

isthe cardinalityof

r.

Now

chooseapoint

(a,

b)

M

suchthata

>

xa.

Thena

>

a,for all

A

</suchthat p

.

IrA-

{A

<

rt

#A},

then

]A

]r/]

and

(b)eA

converges to y in

Y. We

shall showthat

(a,y)

M.

Assumingb

#

y,let

h’=

{A

A’b

<

b};

then

IA’]

Irti

and

(b,),e,,

converges to y in

Y.

Usingthenowfamiliar argument basedon

M

beingclosed and convex,wededuce that

(a,

y)

M.

This argumentcanagain be extended to show

[a,

F)

x

{y}

C_

M,

where

[a,

F)

andconsequently

"

x

.

LEMMA

5.5.

Let

27

beanincreasingsingularityon

X

of order

>

0,and lety

Y

have right

order

.

If isamaximal c-filteron

XY

finer than

.7x’l)r(y),

then for all

f

CI(XY), f(.M)

and

f(27

)

converge tothesamelimit in

[0, 1].

PROOF. Assume

is

/

CI’(X

Y)

and #,i

_>

r

r(y)

suchthat

f(

)

converges to

[0, I]

and

converges tosomepoint

n

[0, I]

other than

.

Snce

#

x

,

t

follows that x

().

We

shallobtaina contradictionbyconstructinga mmlc-filter

x

()

such

that,

for each

L

,

f(L)

ntersects

everyneghbhorhoodofa

n

[0,

I].

It

follows that

f()

converges to

n

[0,

I]

and

hence,

by

Lemma

5.2,

()

converges to

,

acontradiction.

Let

{U

"m

}

be a nestedneighborhoodbe for

[0,

I],

where each

s

a closed nterval.

Snce

y(

x

)

converges toa and

f

>

,

we canfind

<

suchthat

[z,

>

d

[x,>

{}

C,

where

C

{U’

N}

is ac-setin

X

r.

Let

A

{A’p

A <

}.

Let

()<

beastrictly decreeingnetconvergingto in

.

For

each

A

A,

thereis p

<

f

such that

(z,z)

C,

for all z

[,].

Choose astrictly decreeingnet

()

such that

,

[,)

for all

A

A;

then

(y,)

converges toy and

(z,,)

isa net in C.

Let

be the filter of

sections ofthe net

(z,9,),

andlet be y mimal c-filter finerth

I()

O().

SinceC isac-set, C and hence C

.

Thus

f()

converges to a in

[0,

1].

Since each set of the form

[z,

Y>

x

[y, y],

for

<

f,

containsanelementofthe net

(z, 9),

is finerthan

the c-filter

Y

x

(y),

andconsequently

Y

x

().

To

completetheproof, it remains toshowthat

#

Y

x

,

andtherefore that

Y

x

(y).

Y

x

,

then each S contains a set of the form

F

x

{y}

for some

F

Let

S

K

,

where

K

h the form

K

{(z,y)

A

}

for some A. Ifa

set of the form

F

x

{y}

K

for some

F

Y,

then there is an ordinal such that

[z,>

F;

thus

[z,Y)

{y}

K

.

Let be any ordinal such than

<

<

f.

Then

O

{(z,z)

z

z}

{(z,z)

z

y,.}

isac-set in

X

x

Y

containing

K,

d therefore

K

O.

But

[z, z)

x

{y}

O

,

dso

[z,

>

x

{y}

K

.

The sumption that

Y

x is hereby contradicted,dtheproofof the lena iscomplete.

THEOREM

5.6. Let

X

d

Y

betotally orderedspies. Then

o(X

x

Y)

oX

x

oY

the following condition

(.)

issatisfied:

(*)

either

X

or

Y

containsanincreing

(or

decreeing)

singulity of order w,thenthe other spacecontainsnostrictly decreeing

(or

strictly

increing)

sequence.

PROOF.

o(X

x

Y)

oX

x

oY,

the proof that

(2)

(3)

inTheorem3.4 establishes condition

(*).

Conversely,sume

(*)

andconsiderthe diagrm

wo(XxY)

o(XxY)

XxY

To

show

o(X

x

Y)

oX

x

oY,

it is sumcienttoshowthat

(’)-’()

is asingleton for each comptification point in

oX

oY.

Two

cesmustbe considered.

CASE

1.

(Y, 9),

where

Y

d

9

esingulitieson

X

d

Y,

rpectively.

Y

d e eitherbothincreingorboth decreeing,itfollows by

Lena

4.2that

(’)-()

isa singleton.

So,

without loss of generality,sumethat

Y

is increingsingulity of order

f

and is a

decreeing singulity of order

.

f

,

then the istenceofa decreeing singulityon

Y

impliesthe existenceofastrictly decreeingsequence in

Y,

contraryto condition

(*).

Thus there

isnoloss ofgeneralityin suming

>

w;

wealsosume,inviewof

(*),

that everystrictly increingsequenceon

X

converges in

X

deverystrictly decreeing sequenceon

Y

converges in

Y.

-l(a),

thenforall

f

CI’(X

x

Y), f(/)

and

f(31)

converge to thesamelimit in

[0,

11.

Thus, byTheorem 3.3,

e(/)

e(N)

in

o(X

x

Y);

inother

words,

(a’)-l(a)

isasingleton.

CASE 2. a

(jr,.),

whereone member of this pair is a singularity and the other a fixed ultrafilter. Without loss of generality, we assume that jr is an increasing singularity of order

_>

o, and

),

where y E

Y

hasleft orderp

_>

0and right order r/_>0.

We

first observe that

Jr

x

)

is a maximal c-filteron

X

x

Y

and obviously

Jr

x

To

complete the proof, it is sufficient to show that if 31 is any maximal c-filter on

X

x

Y

finer than

J"

x

(4(y)C "l)t(y)),

then

f(31)

and/,(jr

x

9)

converge to the samelimit in

[0,

1]

for all

f

CI’(XxY);

then thedesired conclusionthat

(a’)-I (a)

isasingleton

follows,

asin

Case

1,from Theorem3.3.

Furthermore,

since

q4(y)

and

34(y)

areboth c-filterson

Y,

3t

_>

)"x

(4(y)

C

Vt(y))

implieseither

3t

>_

Jr

x

3),(y)

or 31

>_

Jr

x

3)(y).

If3t isamaximal c-filter finer thanjr x

"l)(y),

itfollowsby

Lemma

5.3that

N

jr x

9,

and the conclusion is trivial.

We

thusassume, forthe remainder of theproof,that 3l

>_

jrx

q4(y).

If

r/ 0, then again

31

Jr

x

)

and the proofis complete. Ifr/

_>

w,

we observethat

Y

contains astrictly decreasing sequence; thus by condition

(*),

>

w, andfurthermore everystrictly in-creasing sequenceon

X

must converge. The existence of the increasing singularity

Jr

on

X

also implies that every strictlydecreasingsequenceon

Y

must converge.

We

nowapply

Lemmas

5.4

and5.5. If r,then

31

jrx 9, and if

r

>

w,then

f(31)

and

/(jr

x

9)

havethe same limit in

[0,

11

forall

f

e

CI’(X

x

Y).

Thus,

under allcircumstances,

(a’)-’(a)

is a singleton, and the proofofthetheoremis complete.

For

totally ordered spaces

X

and

Y,

one canshow that if

X

x

Y

is pseudo-compact, then neither

X

nor

Y

hasa singularity oforder w. Thus itfollowsbyTheorem 5.6that if

X x Y

is

pseudo-compact,/oXx

BoY

o(X

x

Y).

The converseis

false,

as we showedat the endof Section4.

[ll

[21

[31

[4]

[5]

[6]

[7]

[8]

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J.,

Ordercompactificationsoftotallyorderedtopologicalspaces,J.Approximation

Theory

13

(1975),

56-65.

CHOE, T.and

PARK, Y.,

Wallman’s type ordercompactification,

Pacific

d.Math.82

(1979),

339-347.

FLETCHER, P. and

LINDGREN, W.,

Quasi-uniformspaces,Lect. Notes inPureandAppl. Math.,

Vol.77, Marcel Dekker,

Inc.,

New York,1982.

GAMON,

M.andKENT, D., Anote onorderedplanes, Internat.d.Math.

&

Math.Sci. 15

(1992),

199-202.

GLICKSBERG,

I.,

Stone-ech

compactificationofproducts, Trans. Amer.Math. Soc.90

(1959),

369-382.

KENT, D.,

On the Wallman ordered compactification,

Pactfic

J. Math. 118

(1985),

159-163.

KENT,

D. and RICHMOND,

T.,

Ordered compactificationsoftotally ordered spaces, Internat. J.Math. &Math.Sci. 11

(1988),

683-694.