THE NACHBIN COMPACTIFICATION VIA CONVERGENCE ORDERED SPACES

We construct the Nachbin compactification for a T3.5-ordered topological ordered space by tailing a quotient of an ordered convergence space compactification. A variation of this quotient construction leads to a compactification functor on the category of T3.5-ordered convergence ordered spaces.

regular convergence ordered space satisfying conditions C and 0 to be a Ts.s-ordered convergence ordered space, and we show that for such a space X, the regular modification r(X*/) of the quotient X'/ is a regular, T2-ordered convergence ordered compactification of X. Relative to this compactification functor, the regular, T2-ordered, compact convergence spaces (with increasing, continuous maps as morphisms) form an epireflective subcategory of the category of all T3.s-ordered convergence ordered spaces (with increasing, continuous maps as morphisms).
We introduce some basic notation and terminology and summarize some results 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; note that 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 increasing hull of Y'; the decreasing hull d(Y') and convex hull Y? are defined analagously.A filter Y" is said to be convex if " f.Note that i(Y') v d(Y').
If (X, <_, -,) is a poser (X, <) equipped with a convergence structure --, which is locally convex (i.e., f z whenever Y" --, z), then (X, <,--,) is called a convergence ordered space; we usually write X rather than (X, <, -,) when there is no danger of ambiguity.A convergence ordered space is Tx -ordered if the sets i(x) and d(x) are closed for all x X, and Tn-ordered if the order <_ is a closed subset of X X.For any convergence ordered space X, let CI(X) (respectively, CD'(X)) denote the set of all continuous, increasing (respectively, decreasing) maps from X into [0,1].
A convergence ordered space whose convergence structure is a topology is called a topological ordered space.Such a space is said to be convex if the open monotone (i.e., increasing or decreasing) sets form a subbase for the topology.For the remainder of this paper, we shall adopt the notational abbreviation used in [4] and write "t.o.s" instead of topological ordered space" and "c.o.s." in place of "convergence ordered space".A t.o.s.X is said to be T3.a-orderedif it satisfies the following conditions: (1) If z 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 all y A; (2) If z y in X, there is f CI'(X) such that f(y) 0 and f(z) 1.The Ts.s-ordered spaces are precisely those which allow T-ordered t.o.s, compactifications, and all Ts.-ordered spaces are convex.
If X is a T.-ordered t.o.s., then the Nachbin compactification of X (see [1], [6]) is obtained by embedding X into an "ordered cube , whose component intervals are indexed by CI'(X).The Nachbin compactification/0X is characterized by the following well-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 map and Y is a compact, T-ordered t.o.s., then f has a unique, increasing, continuous extension f'/0X -, Y.
We next describe briefly the construction of the convergence ordered compactification X' of an arbitrary c.o.s.X described in [4], which has essentially the same lifting property as 0X.
Given a c.o.s.X, let X be the set of all non-convergent maximal convex filters on X, and let X' { z X} X.Before proceeding further, it will be useful to establish the following proposition about maximal convex filters.
PROPOSITION 1.2.The maximal convex filters on a poset X are precisely the set {f Y" is an ultrafilter on X}.
PROOF.Clearly every maximal convex filter is the convex hull of every finer ultrafilter.Conversely, suppose Y" is an ultrafilter on X and is a convex filter such that f <_ ..T hen for any convex set (7 e , the filters '1 and r2 generated by {i((7)f F" F E r} and {dC( 7respectively, are well-defined filters finer than, and hence equal to, r.Thus i( (7) implies i(G) n d(G) G ; therefore .f. Again assuming that X is an arbitrary c.o.s., let o X X' be defined by (x) 5, for all x C X. A partial order <_ is defined on X as follows: " _< iff i(3r) <_ (or, equivalently, d(.) <_ r).Since x _< y iff <_ , '(X, <_) (X , <_) is an order embedding.
If A C_ X, let A' ()" E X A C Y'); if Y" C F(X), let r denote the filter in F(X) generated by (F' F C Y').A convergence structure on (X , <_) is defined as follows: For C F(X), *-, Eo(X) itf there is Y -, z such that Y"_<4; Writing X' in place of (X', _<', -,), we state the following result which is proved in [4].PROPOSITION 1.3.If X is a c.o.s., then (X', ) is a convergence ordered compactification of X.If f X Y is a continuous, increasing map and Y a compact, regular, T2-ordered c.o.s., then f has a unique, increasing, continuous extension f, X" --I/'.
Recall that a convergence space Y is regular if clr " x whenever x.Here "c/r" is the closure operator for Y, and citY: is the filter on Y generated by (clyF" F ').
Let Y be the filter on X generated by sets of the form F {(a, b) X -< a < 0, b 0} for each natural number n, and let x (0, 0).Let .b e the convex hull of any ultrafilter containing the set S ((a, b) E X a -b-1) and coarser than the filter generated by sets of the form H, ((a,b) _ X" b >_ n) for n 1,2,3,....Note that ($1) is violated by 3r,. and x; thus the compactification X" of X is not T2-ordered.
2. 0X AS A QUOTIENT OF X'.
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 there is, for any f CI(X), a unique, continuous, increasing extension fo X" --, [0,1].
We also impose on X/] the quotient convergence structure which is described (see [2]) ] and there is a filter { E F(X) such that { *-Y in X and ({) _< .
PROOF.X'/, is obviously compact.To show that X'/P. is T-ordered, it is sufficient (by Proposition 1.2, [4]) to show that if , (9 E F(X'/), [F] and (9 --, [] in X'/,, and has a trace on the order _, 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 is well-defined and f e CI'(X'/).If [Y'] and O [.] in X'/, and has a trace on _<, it follows that f() f(O) has a trace on the order of [0,1]; since [0,1] is T2-ordered, f([']) f,(.T) _ f,() f([.]).The latter inequality holds for all f e CI'(X), and so [,v] _ [], which establishes that X'/. is T2-ordered.For an arbitrary c.o.s.X, we have already defined the continuous, increasing maps o X --, X' and a X" X'/; we define o X X'/ by o a o o.It is clear that o(X) is dense in the compact, T2-ordered c.o.s.X'/.We are now interested in characterizing those spaces X for which (X'/.,,) is a compactification.With this goal in mind, we introduce the following conditions.
CONDITION C. For any maximal convex filter Y" on X, r x in X iff/() .f(x) in [0,1] for all f CI(X).
CONDITION O.For any points x,y in X, x _< y in X iff f(x) <_ f(y) in [0,1], for all f CI'(X).
It is easy to verify that any Ta.-ordered t.o.s, satisfies Conditions C and O.
LEMMA 2.2.ff X is a c.o.s, satisfying Conditions C and O, then [] (k), for all x X. PROOF.CI*(X) separates points in X by Condition O, and so a is one-to-one on (X).This implies [] if x y.Next, assume that there is Y" X [].Then f.(Y') f,() f(x) for all f CI(X); in other words, f(,) f(x) in R, for all f CI*(X).Condition C then implies :T x in X, contradicting the assumption :T X'.THEOREM 2.3.Let X be a c.o.s.Then X --, X'/ is an order and a homeomorphic embedding iff X satisfies Conditions C and O.
PROOF.Suppose that X satisfies Conditions C and O. Then is one-to-one since CI'(X) separates points in X.Also note that a.o (a]{x)) o o, and thus al(x is one-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 which implies ,f(x) <_ f(y), for all f e CI(X).By Condition O, z <_ y.Thus o is increasing, and we conclude that o is an order and homeomorphic embedding.Conversely, assume that o is both an order and homeomorphic embedding.Let Y" be a maximal convex filter on X such that, for some e X, ]'() ]'(x) for all f e CI'(X).Suppose not true.Then we need to consider two cases.CASE 1. Y" y and yx.This implies that for each f CI*(X), y(Y') y(y).From this we deduce that [] [], which is a contradiction, since p is assumed to be one-to-one.CASE 2. Y X.This leads to the conclusion that [Y'] [3]; in other words, p(Y) [3] in X/, which implies Y" x in X, since is a homeomorphic embedding.This contradicts Y E X'.We therefore conclude that X satisfies Condition C. Finally, let x,y X such that f(x) < y(y) for all 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 an order embedding.Therefore, X satisfies Condition O. THEOREM 2.4.For every c.o.s.X satisfying Conditions C and O, ((X/), ) is a T2-ordered c.o.s, compactification of X.Furthermore, for any compact, regular, T2-ordered c.o.s.Y and for any continuous, increasing map f X Y, there is a unique, continuous, increasing extension fp.:X'/ r.PROOF.The first assertion is an immediate corollary of Theorem 2.3.The second follows easily with the help of Proposition 1.3.
For any c.o.s.X, let c#oX be the t.o.s, consisting of the poser (X, <) with the weak topology induced by GI'(X).Note that GI'(X) PROPOSITION 2.5.Let X be a c.o.s, satisfying Condition G. Let X oX be the identity map.Then is an order isomorphism and a homeomorphism relative to ultrafilter convergence.
PROOF.It is obvious that is a continuous order isomorphism.Let " --.x in 0X, where F is an ultrafilter.By Proposition 1.2, is a maximal convex filter and f(F)f(z) implies f(]) f(x) in [0,1], for all f E CI'(X).Condition C thus guarantees thatx in X, and hence r z in X.
PROPOSITION 2.6.If X is a c.o.s, satisfying Conditions G and O, then c#oX is a T3.s-ordered t,o.s.
PROOF.First observe that coX also satisfies Condition C and O; O is obvious, and C fol- lows from Proposition 2.5, since and C#oX have the same ultrafilter convergence and hence, by Proposition 1.2, the same convergence of maximal convex filters.
For / E C'I*(oX), let !be the closed interval [0,1] indexed by /, and let P C(X) be equipped with the usual product order and product topology.Then P is a compact, T-ordered t.o.s.Define o CoX P by o(X) , where : C(C#oX) [0,1] is given by (/) /(x), for all / C'I*(C#oX).Since C#oX has the weak topology induced by CI*(CoX) C*(), and C*(o) separates points in C#oX by Condition O, o is a topological embedding (see 8.12, [10]).By Condition O, o is also an order embedding.
Given a c.o.s.X satisfying C and O, we introduce some additional functional notation.Let be the evaluation embedding of the T3.s-ordered t.o.s, c#oX into its Nachbin compactification   and let e eo-i X -/o@oX).The unique extension of e to X' (guaranteed by Proposition 1.3) is denoted by e., and the extension of e 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) are denoted by f, and f', respectively.The following commutative diagram is helpful in keeping track of these various maps. x x.
x./ oX o THEOREM 2.?.If X is any c.o.s, satisfying G' and O, then e is an order isomorphism and a homeomorphism relative 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 follows from Theorem 2.4 that e is continuous and increasing.Finally, if is an X*/ since the latter space is compact.It follows by uniqueness of filter limits in both spaces and the continuity of e that el(a) a.
If X is any convergence space, let AX denote its topological modification (i.e., X is the set IXI equipped with the finest topological structure coarser than X.)If X is a c.o.s, satisfying C and O, we obtain from Proposition 2.5 and Theorem 2.7 that X oX and (X'/)) is a compact, T2-ordered t.o.s, homeomorphic and order isomorphic under e to o(taoX).Let Oo woX --X/ be defined by Oo COROLLARY 2.8.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-of X.
One question which deserves clarification is the status of X'/ as a "quotient" of X'.We have indeed equipped X'/R with the quotient convergence structure, but can we interpret _ as the "quotient order" relative to the order _' defined on X'? Various notions of "quotient order" have been considered (for instance, see [5] and [8]), but the order _ is generally different than these.
Instead of regarding the order and convergence structures of X'/ separately, we think that it is appropriate to consider the notion of a "quotient c.o.s.",where order and convergence structures are considered together.From this perspective, the next theorem indicates that X'/R is indeed a quotient c.o.s, of X', at least in the category of c.o.s.'s which satisfy Conditions C and O. THEOREM 2.10.For a c.o.s.X, let X" and X'/R be defined as before.Let Y be any c.o.s.satisfying C and O, and let h" X'/] -Y.Then h is continuous and increasing iff h o X" Y is continuous and increasing.
PROOF.If h is continuous and increasing, the same is obviously true for h o .
Conversely, suppose h o # is continuous and increasing.Let q --[Y'] in X'/R; then there is ' E [Y'] and a filter/ on X" such that/{ yt in X" and _ r(A).Hence h o r(A) ---, h o (.7)   in Y, by continuity of h o r.But @ _ r(A) and r(r) [r], so h(@) --, h([.]), implying that h is continuous.
To show that h is increasing, let ey be the natural map from Y into o(woY) and consider g er o h o o o X -o(taoY).Since g woX --* o(caoY) is also continuous and increasing, there is a continuous, increasing extension g" o(taoX) (wY) which makes the diagram below commute.Thus erohoo g" oeoo, d since o X X'/ is a dense iection, eoh g" o.
But er order embedding, so h e o g" o e, d h increg.In this brief concluding section, we introduce the notion of a Ta.s-ordered c.o.s., describe the largest regular, T2-ordered c.o.s, compactification of such a space, and interpret this compactifi- cation in the language of category theory.The necessary categorical terminology can be found in In [3], a convergence space X is defined to be completely regular if it allows a symmetric corn-