ORDERED CAUCHY SPACES

This paper is concerned with the notion of ordered Cauchy space which is given a simple internal characterization in Section 2. It gives a discription of the category of ordered Cauchy spaces which have ordered completions, and a construction of the fine completion functor on this category. Sections 4 through 6 deals with certain classes of ordered Cauchy spaces which have ordered completions; examples are given which show that the fine completion does not preserve such properties as uniformizability, regularity, or total boundedness. From these results, it is evident that a further study of ordered Cauchy completions is needed.

Let X be a set, and F(X) the set of all (proper) filters on X.If -,8 E F(X), and F N G # for all F E , G then D denotes the filter on X generated by {F N G: F G E }.On the other hand, if F N G for some F and some G , we say that U fails to exist.
If A c X, A , we denote by <A> the filter of all oversets of A; one departure from this convention are the fixed ultrafilters, which are denoted by for xEX.
Turning to the product set X x X, the diagonal {(x, x): x E X} is designated by A. Compositions are defined as follows: for subsets A, B of X x X, A B {(x, y)   there is z X such that (x, z) E A and (z, y) B}.Compo- sitions and inverses of filters on X x X are defined in the obvious way.If A c X x X and H c X, then A(H) {y E H: (x, y) A for some x H}; if, in addition, G E F(X x X) and F(X) then A(), (H), and () denote the filters on X generated by the sets {A(H): H }, {A(H): A }, and {A(H): A 6 , H }, respectively.DEFINITION I.I.A set of filters on X x X is called a quasi-uniform convergence structure on X if: (u) (u 2) .Q E 0 whenever 0 (u 3) , whenever 5, If satisfies the additional condition (u 5) , o implies -I E o then is called a uniform conver.gencestructure on X.
Quasi-uniform convergence structures are natural generalizations of quasi- uniformities.Uniform convergence structures were introduced by Cook and Fischer [3] another term often used for the same concept is "pseudouniforme Struktur" (see [6]).
A subset ' of a quasi-uniform convergence structure @ is called a base for @ if, for each , there is ' such that c One convenient for any quasi-uniform convergence structure is A : c__ < A >}.
The set Qu(X) of all quasi-uniform convergence structures on X formq complete lattice under the natural ordering (the dual of set inclusion).If 0, Qu(X), then O ^has for a base all finite compositions I n -I where for each j, j E O A or j A" If o E (X), then v is the uniform convergence structure on X induced by o.The lattice operations "v" "^" applied to quasi-uniform convergence structures (including uniform convergence structures) will always be taken relative to the lattice Qu(X).
DEFINITION 1.2.A set of filters on X is a Cauchy structure on X if: (c I) x C for all With every uniform convergence structure on X, there is associated the induced Cauchy structure E F(X): x }.On the other hand, one can associate with each Cauchy structure a finest compatible uniform convergence with base consisting of all finite intersections of filters of the form x , where (one can show that this set of finite intersections is closed under finite compositions).Also associated with a Cauchy structure C on X is the convergence structure q defined by: x in (X, qc iff x E If f: (XI, ) (X2, 2) is a function from one Cauchy space to another, then f is Caucy-continuous if f()

C2 whenever
We next consider the notation and terminology associated with an order relation on X.We shall always assume that (X, is a poset, and we identify the order relation with its graph {(x, y) x y} X x X.Indeed, it will usually be convenient to designate the poset by (X,) rather than the more conventional notation (X, ).Let (X, ) be a poset" let x X, A X, and F(X).
({x}), (A), and () are the increasing hulls of x, A, and respectively.The decreasing hulls are -I (x), -i (5), respectively.The convex hull of A is A^(A) -I(A), and A is convex if A A^similar is increasing if f( I 2 (equivalently, if f(xl) f(x 2) whenever x E x2.
Throughout this section, (X, ) will be arbitrary poset.
DEFINITION 2.1.Let (X, be a uniform convergence space.Then (X, , is an ordered uniform convergence space if there is a quasi-uniform convergence structure on X such that <> , V -I, and = D{: @}. DEFINITION 2.2.Let (X,) be a Cauchy space.Then (X, , is an ordered Cauchy space (abbreviated o.c.s.)If there is a uniform convergence structure on X such that (X, , ) s an ordered uniform convergence space, and C C We see that (X, @, ) is an o.c.s, iff and are both determined by some quasi-uniform convergence structure on X (in the sense described in the preceding definitions).We shall obtain a more precise chmracterization of of this concept by making use of a particular quasi-uniform convergence structure o , con- structed directly from and .Let o= F(X x X): <> be the quasi-uniform convergence structure on X generated by .If is a Cauchy structure on (X, ), we define ,C ^we also define the associated -I uniform convergence structure ,C (,C) v (,) LE4A 2.3.Let (X, ) be a poser.For arbitrary filters , on X, the following statements are equivalent: (a) O() U exists.
PROOF: Assume $ < in C. ,C contains all filters of the indicated form.Furthermore, o has a base of filters of the form <> I < > < > < >, ,C n where each 6j is a basis filter for C that is, a finite intersection of filters of the form x where 6

C
A straightforward set-theoretic argument shows that all such basic filters for @,C can be expressed in the form specified in the statement of the proposition.
PROPOSITION 2.5.Let (X, ) be a poser, C a Cauchy structure on If (x) U <@ > exists, then is clear.Otherwise, for some j, ( x) U(I() x ()) exists.This implies < .j< .3< and the proof is complete.| PROPOSITION 2.6.Let (X, @) be a poset and C a Cauchy structure on X.
Then (X, , C) is an o.c.s, iff U{ 6: C and ,C is compatible with C-PROOF.If the two conditions are satisfied, then (X, , C) is an o.c.s, by Definitions 2.1 and 2.2.
Conversely, assume that (X, , C) is an o.c.s., and let be as in A poser with Cauchy structure is defined to be T2-ordered if x implies x y.This terminology coincides with the definition of "T2-ordered' for convergence ordered spaces in [8] if we identify a convergence structure as a complete Cauchy structure.It foiiows from Proposition 1.2, [8] that if (X, ,) is a T2-ordered poser with Cauchy structure, then is closed in X x X.
A poser with Cauchy structure (X, , C) which is locally convex is said to satisfy Condition (0C) If .C whenever < and < , then (X, , C) is said to satisfy Condition (0C)2 Finally, if (X, ), C) is T2-ordered then we say that Condition (OC) 3 is satisfied.
PROPOSITION 2.8.An ordered Cauchy space satisfies conditions (OC) I, (OC) 2, and (OC) 3 PROOF.Let (X, C be an ordered Cauchy space.Condition (OC) 3 is an immediate consequence of Propositions 2.6 and 2.7.Proposition 2.6 also asserts that ,C is a compatible with C this fact will be used to establish (OC)   and (OC) , n. c. | THEOREM 2.9.Let (X, @ be a poser with Cauchy structure.Then (X, ,C is an o.c.s, iff Conditions (OC) I, (OC2), and (OC) 3 are satisfied.
PROOF.In view of Propositions 2.6, 2.7, and 2.8, it remains only to show that C is compatible with O,C when the three conditions are satisfied.In other words, we must show that B x B o, C implies

C
We prove this implication first under the assumption that is a free filter; later, this restriction will be removed.and If x ,C then by Proposition 2.5 there are filters I' n I n in , with j ~< j for j n such that x _ ([ -i(j) x 8(j): j n}) <>.If K is an ultrafilter finer than then either < > K x contrary to our assumption that is free, or else there is an index j, j n such that -I (j) x (j) x K.In the latter case we see easily that '3 ~< '3' and Sj ~< j; thus by (OC) and (OC)2, j j Sj)^C Without loss of generality, let {I,..., m} (m <_ n) be the indices for which, if j {I m }, there is an ultrafilter K finer than such that (j) x (j) c E x K For each j, _< j _< m, let j 5.
.)^as in the preceding paragraph, Z. C We next show that ={.j=l m} C Indeed, suppose j, k {I,..., m}.Then there are ultrafilters Kj and k finer than such that Zj c j and Zk c .Since K x K D it follows that either Aj x ) U < > exists, or else (Kj x K k) U )-I() x ()] exists for some ,C is not free, then by (OC) 3) is necessarily of the form I where I is free and x X.By our previous results, I C Also, 6 c_ x where is the filter described earlier in the proof.Thus c_ x ]I implies <> U ( x I) exists or ()-l(j) x 9(j))U( x | exists for some j, j -< n.Either way, it follows that < I Starting with 6 c__ I x x leads to the conclusion that i < Using (0C)2 again, we con- C, and the proof is complete.
clude that

I
The three conditions which characterize ordered Cauchy spaces are all quite natural.The properties "locally convex" and "T2-ordered" are commonly assumed in the study of ordered topological and convergence spaces, and the condition (OC) 2 is a natural extension to ordered spaces of axiom (C 3) of Definition 1.2.Cauchy structures intrinsic to distributive lattices and lattice ordered groups studied by Ball ([I], [2]) are examples of ordered convergence structures.
Since Cauchy structures are primarily used as a means for constructing com- pletions and compactifications, it is natural to turn our attention now to completions of ordered Cauchy spaces.While it is well known that all T 2 Cauchy spaces have T 2 completions (indeed, a variety of different completions are described in the literature), it turns out that not all ordered Cauchy spaces have ordered Cauchy completions.In the remainder of this paper, we characterize those ordered Cauchy spaces which do have such completions, and examine several special cases.

ORDERED CAUCHY COMPLETIONS.
A triple (X, , C) will be called a poser with T2-ordered Cauchy structure if (X, ) is a poset and C a T2-ordered Cauchy structure on (X, ).Thus a poset with T 2 ordered Cauchy structure is required to satisfy condition (OC) 3 but not (OC) or (OC) 2.An increasing, Cauchy-continuous function from one such space to another will be called a morphism.The category having posets with T2-ordered Cauchy structures as objects and morphisms as maps will be designated PCS.Let OCS be the full subcategory of PCS whose objects are the ordered Cauchy spaces.Cauchy modification of C relative to (X, )).If f: (X, ), C) (X I, I' C1 is a map in PCS, then it is a simple matter to verify that f:(X, is also a morphlsm. PROPOSITION 3.1.OCS is a reflective subcategory of PCS relative to the reflector F: PCS OCS defined for objects by F(X, , ) (X, ),)C) and fixed on maps.
If an object (X, ,C) in PCS is complete, then we shall refer to C as a convergence structure on X and (X, C as a convergence, s.pac.For complete objects in OCS, the T2-ordered property means simply that is closed in the product convergence on X x X.A complete o.c.s, will be called an ordered cpnvergence space; the latter term has the same meaning as "T2-ordered convergence ordered pace" in [8].Let OCON be the full subcategory of OCS whose objects are ordered convergence spaces. DEFINITION 3.2.An ordered completion ((X I, I,CI), ) of an o.c.s.
(X, , ) is a complete o.c.s.(XI' I' CI) and a morphlsm @:(X, , ) (X I, O I, i which is an ordered Cauchy embedding (meaning that and -I one-to-one morphisms and (X) is dense in X I). are The maln goal of this section is to determine which objects in OCS have ordered completions; the full subcategory of OCS determined by these will be denoted by COCS.It is clear that (X, 9, C) COCS iff (X, , ) is isomorphic to a Cauchy subspace of an object in OCON; we seek, however, an internal characterization of such objects.For this purpose, it is necessary to introduce a new order relation on the Cauchy filters of an o.c.s.
Let (X, , C) be an o.c.s.For C let L {x X: x and U x X: x}.Note that if C then LN -l(y) and U (y).Now, for , C define iff any one of the following is true: , or v < L> exists, or v < U > exists.Let @ be the smallest transitive subset of Cx C containing other words, iff there are I''''' n in C such that I n " Note that I 2 C implies I O 2 and 2 i also, if then I O I for every I [] and I ]" Before proceeding further, we give an example to illustrate the difference between the relations and Q EXA}LE 3.3.Let X R 2 {(0,0)} be the Euclidean plane with the usual order and with the origin deleted.Let D(x) be the neighborhood filter at x with respect to the usual topology.Let be the filter on X generated by all sets of the form Gn {(x,0): 0 < x < in for n 6 N and let be generated by nil If (Xl' I' CI)' ) is an ordered completion of (X, , C), then ((X I, I ) is obviously a Cauchy completion of (X, C), and so it will be con- venient to review some aspects of Cauchy completions.Starting with a T 2 Cauchy space (X, C), two filters , in C are equivalent if .Let X*

{[]:
C} be the set of all Cauchy equivalence classes; let X X* be the natural injection, defined by j(x) [].The completion ((X I, CI), ) is in standard form if X X* and j.Reed [i0] showed that every Cauchy completion is equivalent (in the usual sense) to one in standard form.The same is, of course, true for ordered completions.sets of the form H {(0,y): 0 < y < i }.Let C be the smallest Cauchy structure on X which contains D(x) for each x in X along with , and .It is easy to verify that (X, , C) is an o.c.s.; also note that B and B i .
However '$ and U both consist of the closed first quadrant (excluding, of course, the origin), and consequently Q and We now introduce three conditions which are analogous to (OC) I, (OC)2, and (OC) 3" we shall show that an o.c.s. (X, @, ) has an ordered completion iff it satisfies these new conditions. (COC) C,,Q, and O implies 3 C implies x _< y.
(3) (X, , C) has an order-strict ordered completion. (4)(X* O* *) is an ordered completion of (X , ) Without repeating the relevant definitions here, we remark that, in the terminology of [5], the functor K: COCS OCON, defined by K(X, , C) (X*, *, C*) is an order-strict completion functor, and consequently that COCS is a completion subcategory of OCS.We shall henceforth refer to ((X*, @*, C*), j) as the fine ordered completion of (X, , C), and K will be called the fine ordered completion functor. 4. UNIFOMIZABLE ORDERED CAUCHY SPACES.
It is natural to ask whether the fine completion functor K, when restricted to uniformizable o.c.s.'s, preserves uniformizability.As the next example shows, the answer is no.The same example shows that the fine completion preserves neither regularity nor total boundedness.EXAMPLE 4.4.Let E be the Euclidean plane with the usual topology and partial order: E {(a,b), (c,d)): a c and b _< d}.Note that the elements in E are E-related iff they lie on the same vertical line.Let Y {L n N} U n {(0,0)}, where Ln {( y): 0 y _< }; let (Y, O', 4') be the compact, T zordered topological space (considered as an object in OCON) with order and topology inherited from E. Let S {( n N} and let X Y-S.The order and Cauchy structures which X inherits from Y are denoted by O and respectively.Since (Y, 0', ') is a compact, T2-ordered completion of (X, O, ), (X, , ) is a uniformizable o.c.s.Now consider the fine completion ((X*, 0., *), j) of (X, O, ).There is an obvious correspondence between the sets X* and Y relative to which the equivalence classes of non-convergent filters in X* correspond te the subset S of Y. Let be the Frechet filter on Y of the sequence ( ), and let be the corresponding filter on X*.Although converges to (0,0) in (Y, 4'), is non-convergent relative to (X*, *).If p ( X* is the equivalence class of filters converging in (X, ) to the origin, then the closure of the p-neighborhood filter in (X*, *) is nonconvergent, and so (X*, *) is not regular, and consequently not uniformizable.
From the fact that (Y, ', ') is compact, it also follows that (X, , ) is totally bounded (meaning that every ultrafilter on X is in ).Since no ultrafilter finer than converges in (X*, *), we see that total boundedness is not preserved by the fine completion functor.
Although no definition of "regularity" has been given for ordered Cauchy spaces, the space (X, O, 4) as an ordered Cauchy subspace of a T2-ordered, compact topological space, would be "regular" by any reasonable definition of that term.Since (X*, G*, ) is not regular (in the usual convergence space sense) no reasonable notion of regularity is preserved by the fine completion functor.

TOTALLY ORDERED CAUCHY SPACES.
We begin with a simple condition which is sufficient for the existence of an ordered completion of an o.c.s.DEFINITION 5.1.An o.c.s.(X, , ) satisfies Condition (A) if, whenever is nonconvergent, there is ] such that O -I (I) c__ < L > and i i < U, > then: PROPOSITION 5.2.If (X, @, g) is an o.c.s, which satisfies Condition (A), (I) The relations Q and < on coincide.
PROOF.(I) Suppose U < L > exists.If converges to y, then L -l(y), and < follows intmediately.If is nonconvergent, Condition (A) guarantees that U -I() exists, and so < Similar reasoning shows that < whenever U < U > exists.Thus it is clear that Q implies < The converse is always true.
The term totally ordered Cauchy space (abbreviated t.o.c.s.) will be used for any o.c.s.(X, ,C ) for which @ is a total order.PROPOSITION 5.3.If and are filters on a totally ordered set (X, @), then either (5)!J exists, or -1() U exists.
PROOF.If both fail to exist, there is F E and G such that (F) 0 G and -I(F) G , which means that ((F) U )-I(F)) 0 G .
PROOF.The existence of an ordered completion follows immediately from Proposition 5.2 and 5.5.The completion (X*, *, C*) is totally ordered by Proposition 5.3 and the remark following Proposition 5.2.Since the total order @* is the smallest allowable order for an ordered completion in standard form, it is the only possible order for an ordered completion in standard form.

AN ORDERED COACTIFICATION.
Every T2-ordered compactification of an ordered topological or convergence space can be regarded as the completion of a certain totally bounded o.c.s., and so the entire subject of T2-ordered compactifications lies within the scope of our present investigation.Our immediate goal, however, is rather modest; we shall formulate the ordered convergence compactification constructed in [8] as an ordered Cauchy space completion, thereby gaining some further insight into its properties.
Let (X, , +) be an ordered convergence space (i.e., an object in OCON); the notation "7 x" indicates that converges to x in this space.Let C be the complete Cauchy structure on X consisting of all convergent filters relative to and (following [8]) let X' be the set of all nonconvergent, maximal convex filters on (X, O, +).In [8] an ordered convergence space (X, O, +) is defined to be strongly T2-ordered, if the following property and its dual are satisfied: If x, X' and () exists,

-I() c
With each ordered convergence space (X, @, +), we associate the Cauchy structure C U F(X): there is 6 X' such that G c }.Note that is the finest Cauchy structure compatible with which is both totally

7 .
If (X, ) is a poset and C a Cauchy structure on X, from these observations.
{I n }, where 5 < .Either alternative leads to the < Applying the same" reasoning with indices reversed leads conclusion that j k < Z and by (0C)2, j k to conclusion Zj k Z C. Since j and k are arbitrary indices in {l,...,m}, If (X, , C) PCS, let C be the Cauchy structure compatible with the -I uniform convergence structure ,C o ,C v o,C It follows from Proposition 2.7 that (X, , @C) is T2-ordered.Furthermore, ,C ,C and so by Proposition 2.6, (X, , is an o.c.s.We shall call C The ordered proof is concluded by observing that if D < L > or U < U > exists, then([,], []) is in the closure of Since )I is both closed and transitive, ([3], ]) * impliesPROPOSITION 3.5.If (X, , ) has an ordered Cauchy completion, then (X, , C) satisfies (COC)I, (COC)2, and (COC)3.

PROPOSITION 4 .
1. Let (X, ,) be a uniform ordered space determined by a quasiuniformity g Then the following conditions hold.