ON THE NACHBIN COMPACTIFICATION OF PRODUCTS OF TOTALLY ORDERED SPACES

Necessary and sufficient conditions are given for o(X Y) oX x oY, where X and Y are totally ordered spaces and oX denotes the Nachbin (or Stone-ech ordered) compactificatio of X.

Our solution to the aforementioned problem makes extensive use of the Wallman ordered compactification woX, introduced in [2].In Section 3 we show that for any T3.-orderedspace X, oX can be obtained from woX via a certain quotient construction, and this result is employed in the proof of our main theorem.We also make use of the fact that woX [oX for any totally ordered space X.
We first prove a preliminary version of our product theorem in Section 4 under the assumption that the totally ordered spaces X and Y are %trictly first countable".(A totally ordered space is defined to be strictly first countable if every neighborhood filter and every maximal closed-convex filter has a countable filter base.)Surprisingly, the condition which works" in the strictly first countable case also works" in the general case (see Theorems 4.4 and 5.6).If X and Y are strictly first countable, then o(X x Y) [oX [oY and Wo(X Y) woX woY are equivalent statements.We do not know if this equivalence holds for arbitrary totally ordered spaces X and Y.
Let {X,_<) be aposet.Fora non-empty subset A of X, we define d(A) {y e X y < x for some x E A} to be the decreasing hull of A; the increasing hull i(A) is defined dually.We shall write 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}}; a set which is either increasing or decreasing is said to be monotone.For any A C_ X, A ^i(A} N d(A} is called the convex hull of A, and A is convex if A A^.
Let F(X) denote the set of all filters on a set X. We always use the term filter o mean a proper set filter.If .T and .are filters on X such that F n G # , for all F E Y and G C ., then r y denotes the filter generated by (F N G" F e ', G e .);if, on the other hand, contain disjoint sets, we say that " v .[ails to exist.A filter " is ]tee if there is no point common to all the sets in .T. A filter which is not free is said to be fixed; in particular, the symbol will denote the fixed ultrafilter generated by x G X.For any filter Y on X, the filter '^g enerated by sets of the form {F^F E .T} is called the convex hull of '.An ordered terpalogical space, or simply an ordered space, is a triple (X, _<, r), where (X, <) is a poser and r a eanvex topology on (X, <_} (i.e., r is a topology which has a subbase consisting of monotone open ets).Note that every ordered space is locally convex in the sense that every neighborhood filter has a base of convex open sets.When there is no danger of confusion, we refer to the ordered space (X, <, r} simply as X.If X and Y are ordered spaces, a map f X Y is increasing (respectively, decreasing) if x < y in X implies f(x) <_ f(y) (respectively,   in Y.A continuous, increasing map is called an ordered topological morphism, or more briefly a morphisrn.A b]jective morphism whose inverse is also a morphism is called an ordered $opological isomorphism, or more briefly an isomorphism.Let VI'(X) (respectively, CD'(X)) denote the set of all morphisms (respectively, continuous, decreasing maps} from an ordered space X into [0, 1].
An ordered space X is T-ordercd if i(x} and d(x} are closed sets, for all x X; X is T-ardered if the partial order relation _< is closed in X x X.An ordered space X is T.s-ordered (completely regular ordered in [8]) if the following conditions are satisfied: (1} If x X, A is a closed subset of X, and x A, then there is f CI'(X) and g CD'(X) such that f(x) g(x} 0 and f(y) v g(y) 1, for all y E A; (2) If x y in X, there is f CI'(X) such that /(y} 0 and f(x) 1.The T3.-ordered spaces are precisely those which allow T-ordered compactifications (see [3] and [8]}.An ordered space X is normally ordered (see [8]} if, whenever A and B are disjoint closed sets, with A increasing and B decreasing, there are disjoint open sets U and V, with b increasing and V decreasing, such that A C_ U and B C_ V.An ordered space which is both normally ordered and Tl-ordered is said to be T4-ordered. Given a Ts.s-ordered space X, there is a largest T2-ordered compactification of X called the Naehbin (or Stone-ech ordered) cornpaetification, denoted by [3oX.The standard construction of /9oX involves embedding X in the "ordered cube" [0, 1] ct'(x) where the latter space has the usual product order and topology.This compactification is characterized by the following well-known exnion orem Oee [] or []).THEOREM 2.1.If X is a Ts.s-ordered space, Y is a compact, T2-ordered space, and f X Y is a morphism, then there is a unique morphism j-r ./oXy such that the diagram X ex \ Y commutes, where ex X --/3oX is the canonical embedding.
We next review the construction of the Wallman ordered compactification.Let X be an oderd p=e ana X; e () (repey, ()) b h ml dod, dr=.g (re- spectively, closed, increasing) set that contains A. Let A a I(A) q D(A); if A A a then A is called a e-set.One may verify that the collection of all e-sets on X is closed under arbitrary intersections.It is obvious that e-sets are dosed and convex, but not all closed, convex subsets of X are e-sets.If jr G F(X), let D(jr) (respectively, I(jr)) denote the filter on X generated by {D(F) F G jr} (respectively, {I(F) F jr}).The filter jra 1(jr) v D(jr) is generated by sets of the form ava, for a v jr; if jr jra, then jr is called a e-filter.One may easily verify (using Zorn's Lernma) that every e-filter is coarser than a maximal e-filter.Let X be a Trordered space, and let WoX be the set of all maximal e-filters on X.A partial order "<" for WoX is defined as follows: jr < = I(jr) C # and D() jr.We also assign to woX the topology with closed subbase g {A A aa}, where A Then woX, with the order and topology just described, is an ordered space which is compact and T (but generally not T-ordered).If x X WoX is defined by x(z) 5, for all z G X, then x is an isomorphic embedding, and consequently (WoX, x) is an ordered eompactification of X.Furthermore, we have the following extension property (see [2] or [6]).
THEOREM 2.2.Let X be a T-ordered space, Y a compact, T2-ordered space, and f X --Y a morphism.Then there is a unique morphism jz.w,X Y such that the diagram An ordered space X is defined to be a c-space if i(A) and d(A) are closed subsets of X when- ever A _C X is a c-set.The next theorem is proved in [6].
3. /oX AS A QUOTIENT OF woX.Throughout this section, X will denote an arbitrary Ts.s-ordered space and (woX,) the Wallman ordered compactification of X.
PROOF.Obviously, #oX is compact.We recall (see [8]) that an ordered space is T-ordered whenever z y, there are disjoint neighborhoods U and V of x and y, respectively, such that U is increasing and V is decreasing.
By Theorem 1.2, there is a unique morphism 'woX oX such that the diagram / x \ oX coutes.Also, for y f CI'(X), there is a unique f CI'(oX) such that f f o e.Note that fA f o , since thee maps agree on the dense subspace (X) of woX, d hence on woX.LEMMA 3.2.For Y', .woX, (Y', ) .(Y') (.).
PROOF.Let e''#oX floX be defined by 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 induced by and " o is continuous, e" is continuous.Since #oX is compac and oX is T, " is a homeomorphism.
4. A PRELIMINARY PRODUCT THEOREM.
Compactifications of totally ordered spaces are studied in [I] and [7], and we begin this section by summarizing some relevent results from [7].We define a totall!/ordered space X to be a T2- ordered space whose partial order is a total order (i.e., if z, /E X, then z <_ /or l/ _< z).It is easy to show that a totally ordered space is a T4-ordered c-space in which the c-sets are precisely the closed, convex sets.Consequently, by Theorem 2.3, the compactifications woX and floX of a totally ordered space X exist and are equal.Furthermore, every T2-ordered compactification of a totally ordered space is itself a totally ordered space.
For a totally ordered space X, we use the equivalence of floX and woX to describe the com- pactification points of [3oX as maximal c-filters.It is shown in [7] that, in a totally ordered space X, the maximal c-filters are precisely the convex hulls of ultrafilters; the non-convergent maximal c-filters on X are called singularities.Given a singularity , let T t3{FT F E 7}, where F denotes the set of upper bounds of F, and yt U{Ft F C }, where F is the set of lower bounds of '.The convex sets and )'t partition X, and so exactly one of these sets is in 7. If 't " (respectively, 't ') we say that .is a decreasing (respectively, increasing) singularit!t.
A totally ordered space X is strictll first countable if every neighborhood filter and every singularity has a countable filter base.If .is an increasing singularity with a countable filter base, then there is a strictly increasing sequence Zl < z < z3 < in X such that is the convex hull of the filter of sections of (z); similarly, each decreasing singularity with a countable filter base is likewise derived from a strictly decreasing sequence 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] that if X is the real line with the usual order and topology and Y is any totally ordered space whose underlying poser is the real line, then X Y is a c-space the topology for Y is the usual topology.Thus, in general, wo(X Y) fails to be Tz-ordered and hence wo(X x Y) and [3o(X Y) are non-equivMent eompaetifieations (see Theorem 2.3).The next two lemmas are due to Margaret A. Gamon.LEMMA 4.1.If X and Y are totally ordered spaces, A a c-set in X, and B a c-set in Y, then A B is a c-set in X Y. LEMMA 4.2.Let X and Y be totally ordered spaces, and let jr be a singularity on X and .asingularity on Y such that either both are increasing singularities or both are decreasing singularities.Then jr . is a maximal c-filter on 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 be the restrictions of jr and .to X and Y, respectively.It is easy to verify that jr and .are increasing singularities on X' and Y', respectively, and that (jr,}r (06'}r 0. It is also easy to verify that jr .is a maximal c-filter on X Y == jrw 6 is a maximal c-filter on X Y.
Therefore we shall 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 .F or anyeEXandd Ysuchthat a<c and b < d, we must have (c, d) C H, since otherwise i(c) i(d) would be a member of jr .disjoint from H.This implies that i(a) i(b) C_ g.But i(a) i(b) C jr , and so C_ jr .It follows that jr ., and consequently jr is a maximal c-filter on X Y.
In general, the conclusion of Lemma 4.2 is not valid if one of the singularities is increasing and the other decreasing.Indeed, the next lemma establishes that if jr is an increasing singularity on X and , a decreasing singularity on Y, both with countable filter bases, then jr x . is not a maximal c-filter on X x Y. LEMMA 4.3.Let X and Y be totally ordered spaces.Let (x.) be a strictly increasing se- quence in X and (y,,) a strictly decreasing sequence in Y. Let S {(x2r,-l, y2,,_)'rt e N) and T {(xz, yz,.,)'nG N}. Then there is g e CI'(X Y) such that g(i(S)) 1 and g(d(T)) O.
net converging to y, we denote by "l)t(y) the filter on r generated by {[y^,y] A < p}; "lit(y) is called the left neighborhood filter at y, and we set "t(y) in case p 0. Likewise, if y has right order > 0 and (z^)^< is a strictly decreasing net converging to y, the right neighborhood filter V.(y) is generated by {[y,z^] A < }; again we set Vr(y) if 0. Furthermore, if > 0 we denote by );(y) the filter on Y generated by {(y,z^] A < }.Note that )t(y) n Yr(y) is the usual neighborhood filter at y.We shall also need additional interval notation pertaining to singularities.If jr is an increasing singularity on X and z E X is such that $ < jr in woX, we define [z, jr) {a E X z < a in X and h < jr in woX} and (z, jr) [z, jr)\{z}.In ease is a decreasing singularity on Y and y Y is such that .< in woY, let (.,y] {a Y" a < y in Y and .< h in woY} and let (.q,y) (.,y]\{y}.If jr has order and (z^)^< is a strictly increasing net such that (:^)^< converges to jr in woX, then each of the sets {Ix^, jr) A < } and {(z^, jr) A < (} is a filter base for jr.Likewise, if .has order rt and (y^)^<.is a strictly decreasing net in Y such that (0h)^<.converges to .in woY, then each of the sets {(.,U^]A < r/} and {(.,U^)A < r/} are filter bases for .
LEMMA 5.1.Let jr be an increasing singularity on X of order > w, let .be a decreasing sin- gularity on Y of order r/> w, and assume that every strictly increasing sequence on X and every strictly decreasing sequence on Y is convergent.If/ and are maximal c-filters on X Y, both finer than jr x 6, then for all f CI(X x Y), f() and f() converge to the same limit in [0, 1].
PROOF.Suppose there is f CI'(X x Y) such that f(} converges to a, f() converges to b, and a b in [0, 1].Let U and V be disjoint neighborhoods of a and b, respectively, and choose closed sets L E and M such that f(L) _C U and f(M) C_ V.Then, choose the following points in X Y Continuing in this way we obtain sequences (a., b,.,) in L and (c., 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 jr be an increasing singularity of order > w on X, and let y Y have right order r/> w.Assume that every strictly increasing sequence on X and every strictly decreasing sequence on Y is convergent.If and are maximal c-filters on X Y, both finer than jr x )(y), then for all f CI(X Y), f(.) and f(t) converge to the same limit in [0, 1].
PROOF.The proof of Lemma 5.2 is essentially identical to that of Lemma 5.1.LEMMA 5.3.Let be an increasing singularity on X of order > w and let y Y have left order rt :> 0. If is a maximal c-filter on X Y finer than 27 e(y), then .M .T .
PROOF.Let (Yu),< be a strictly increasing net on Y converging to y; thus )t(Y) has a filter base {[Yu,Y] # < r/}.Note that 27 has a filter base of the form {Ix^, 27) A < }, where (x^)^< is a strictly increasing net in X.
If b y there is nothing more to show, so assume the contrary, and choose ordinals p < r/ and r < such that b < yp and a < x,.Then (a, b) < (a, b,p) < (a,a, b,a), and so by convexity of M, (a,b,,) M. Since y, <_ b,.,, (a,y,,) M. Indeed this reasoning implies that (a, y,) M for all < rt such that p < u.Since M is closed, (a, y) M. Using again the convexity of M, we deduce that [a, 27) x {y} C_ M and that [a, 27) ,v.Thus .7 > , and since both are maximal c-filters, equality holds.LEMMA 5.4.Let " be an increasing singularity on X of order _> w, and let y Y have right order rt > w.If rt and is amaximalc-filteronX Y finer than 'x r(y), then ,= x .
PROOF.Let (x)<e be a strictly increasing net in X such that {[x,.v)A < } is a filter base for '.Let (Yv),<, be a strictly decreasing net in Y converging to y such that {[y,y,] < is a filter base for "Vr(y).Let M )[ be a closed, convex set such that M C_ [Xo, .7) [y, yo].CASE 1. rt < .If 0 <_ A < rt, choose (ax,b,) M N ([xA,.T) [y, yA]) such that is strictly increasing in X and (bx)A< is strictly decreasing in 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.For each A < r, choose (a,,b,) M [Xo,') x [y,y,] such that (bx)A<, is strictly decreasing; thus (bx)< n converges to y in Y.For each A < rt, let ux be the least ordinal such that a < x.Note that { A < rt} C_ {p p < }.Considering the net (x,)<, we observe that since each # < , there is some {A A < #} such that the term xa occurs times in the net (x,)<,, where Irtl is the cardinality of r.Now choose a point (a, b) M such that a > xa.Then a > a, for all A </ such that p .
IrA-{A < rt #A}, then ]A ]r/] and (b)eA converges to y in Y.We shall show that (a,y) M. Assuming b # y, let h'= {A A'b < b}; then IA'] Irti and (b,),e,, converges to y in Y.Using the now familiar argument based on M being closed and convex, we deduce that (a, y) M.This argument can again be extended to show [a, F) x {y} C_ M, where [a, F)   and consequently " x .
LEMMA 5.5.Let 27 be an increasing singularity on X of order > 0, and let y Y have right order .I f is amaximal c-filter on XY finer than .7x'l)r(y),then for all f CI(XY), f(.M) and f(27 ) converge to the same limit in [0, 1].
PROOF.Assume " has filter base {[x, :7) ,X < } as in the preceding proof.Suppose there is / CI'(X Y) and #,i _> r r(y) such that f( ) converges to [0, I] and converges to some point n [0, I] other than .Snce # x , t follows that x ().
We shall obtain a contradiction by constructing a mml c-filter x () such that, for each L , f(L) ntersects every neghbhorhood of a n [0, I].It follows that f() converges to n [0, I] and hence, by Lemma 5.2, () converges to , a contradiction.
Let {U "m } be a nested neighborhood be for [0, I], where each s a closed nterval.Snce y( x ) converges to a and f > , we can find < such that [z, > d [x,> {} C, where C {U' N} is a c-set in X r.Let A {A'p A < }.Let ()< be a strictly decreeing net converging to in .F or each A A, there is p < f such that (z,z) C, for all z [,].Choose a strictly decreeing net () such that , [,) for all A A; then (y,) converges to y and (z,,) is a net in C. Let be the filter of sections of the net (z,9,), and let be y mimal c-filter finer th I() O().
Since C is a c-set, C and hence C .T hus f() converges to a in [0, 1].Since each set of the form [z, Y> x [y, y], for < f, contains an element of the net (z, 9), is finer than the c-filter Y x (y), and consequently Y x ().To complete the proof, it remains to show that # Y x , and therefore 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. If a 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,.} is a c-set in X x Y containing K, d therefore K O.
But [z, z) x {y} O , d so [z, > x {y} K.The sumption that Y x is hereby contradicted, d the proof of the lena is complete.
THEOREM 5.6.Let X d Y be totally ordered spies.Then o(X x Y) oX x oY the following condition (.) is satisfied: (*) either X or Y contains an increing (or decreeing) singulity of order w, then the other space contains no strictly decreeing (or strictly increing) sequence.

XxY
To show o(X x Y) oX x oY, it is sumcient to show that (')-'() is a singleton for each comptification point in oX oY.Two ces must be considered.(Y, 9), where Y d 9 e singulities on X d Y, rpectively.Y d e either both increing or both decreeing, it follows by Lena 4.2 that (')-() is a singleton.So, without loss of generality, sume that Y is increing singulity of order f and is a decreeing singulity of order .f , then the istence of a decreeing singulity on Y implies the existence of a strictly decreeing sequence in Y, contrary to condition (*).Thus there is no loss of generality in suming > w; we also sume, in view of (*), that every strictly increing sequence on X converges in X d every strictly decreeing sequence on Y converges in Y.
The preceding observations allow us to conclude, using Lemma 5.1, that if and are in PROOF.It is clear that i(A x B) i(A) x iCB).Since totally ordered spaces are c- spaces, i(A) and d(A) are both closed, and so i(A) I(A) and d(A) D(A).Therefore, 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.