@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 1, 2003 pp. 99–114 On the use of partial orders in uniform spaces Bruce S. Burdick Abstract. We investigate the use of nets indexed by preorders in uniform spaces. Nine different Cauchy conditions and four different convergence conditions yield 36 completeness properties, each of which turns out to be equivalent to a known form of completeness. We also use these preordered nets to characterize the functors θ, λ, and ν, which are associated with these completeness properties. In the case of λ we give an example to show that the analogous characterization with predirected nets does not work. 2000 AMS Classification: 54A20, 54D20, 54E15. Keywords: Locally fine, complete, supercomplete, cofinally complete, para- compact. 1. Introduction We have recently written a paper [3] on completeness properties determined by nets whose directed sets are well ordered; this current paper represents the opposite extreme, namely, the use of preorders as index sets for nets. In the title, we use the term ‘partial order’ after the fashion of Kelley, who in Chapter 2 of his book [14] does not require the antisymmetry property for orderings. His partial orders need only be transitive and reflexive; and his directed sets have these properties plus upper bounds for finite sets. But over the years, the terminology has changed, and we follow it, using ‘preordered’ and ‘predirected’ for these concepts below. Section 2 deals with the completeness properties which arise. Section 3 shows how these preordered nets may be used to give new definitions for functors discussed in the literature. These new definitions have some advantages over the definitions previously known. In the case of the functor ν the new definition is internal rather than external. It does not require the use of the completion of the space. In the case of λ the new definition does not make use of Ginsberg’s 100 Bruce S. Burdick and Isbell’s quasiuniformities, i.e., filters of coverings which may fail to satisfy the star refinement property. 2. Variations on the property of Completeness Definition 2.1. In a uniform space (X,U) a filter F is weakly Cauchy if for every U ∈ U there is a U-small set S ⊆ X which has non-empty intersection with every member of F. A net ξ : D → X is cofinally Cauchy if for every U ∈ U there is a cofinal set C ⊆ D such that ξ[C] is U-small. A space is cofinally complete if every cofinally Cauchy net clusters, or, equivalently, if every weakly Cauchy filter clusters. A filter is stable if for every U ∈ U there is an F ∈F such that for all F ′ ∈F we have F ⊆ U[F ′]. A net ξ : D → X is almost Cauchy if for any U ∈U there is a d ∈ D and a set C of cofinal subsets of D such that for each C ∈ C, ξ[C] is U-small, and for each d′ ∈ D, if d′ ≥ d then d′ ∈ ⋃ C. A space is supercomplete if each almost Cauchy net clusters, or, equivalently, if every stable filter clusters. Recall that, originally, Isbell called a space supercomplete if it had a com- plete hyperspace [12, 13]. That condition is equivalent to the definition given here. This author stated this in [2] before discovering that Isbell mentioned this in the last paragraph of [12]. We note that Császár gave this as an open problem in [4]. In [12] and [13], Isbell uses a notion of nets indexed by preordered sets to characterize supercompleteness. In [2] we obtained a slightly different charac- terization of supercompleteness using these preordered nets. We wish to extend these results here, but because of the many different Cauchy and convergence conditions that are possible we need to develop first a streamlined terminology for all the completeness properties that are generated. To do this we make use of the Alexandroff topology. For a preordered set (P,≤), the Alexandroff topology [1] is the collection of upper sets, that is, those A for which x ∈ A and x ≤ y ⇒ y ∈ A. This is easily seen to be a topology which is generated by all sets of the form ↑x = {y : x ≤ y} for x ∈ P, and in this guise was called the partial order topology in Chapter III of the book [17]. When we refer below to open, dense, open dense, or somewhere dense (= not nowhere dense) subsets of a preordered set, we will be assuming that these terms refer to such sets as determined by the Alexandroff topology. Definition 2.2. Any function from a preordered set to a uniform space will be called a ponet. We say a ponet S : (P,≤) → (X,U) is open (dense, open dense, somewhere dense) Cauchy if for each U ∈ U there is an open (dense, open dense, somewhere dense) set R ⊆ P with S[R] ×S[R] ⊆ U. In addition, a ponet S : (P,≤) → (X,U) satisfies the property open/open dense Cauchy if for each U ∈ U the union of some collection of open sets R ⊆ P such that S[R] × S[R] ⊆ U, is open dense, and we may define somewhere dense/open dense, somewhere dense/open, somewhere dense/dense, and dense/open dense in an analogous manner. Partial orders in uniform spaces 101 Definition 2.3. We say x is an open (dense, open dense, somewhere dense) limit of a ponet S : (P,≤) → (X,U) if for every neighborhood O of x, S−1[O] contains an open (dense, open dense, somewhere dense) set. From these nine Cauchy conditions and four convergence conditions we may define 36 completeness properties in the obvious way. For example, the property open/open dense—dense completeness would say that every open/open dense Cauchy ponet has a dense limit. These 36 properties are not distinct, however, as we will show below. The next two propositions are the only previous results we know about this type of completeness. Proposition 2.4 (Isbell [12], equivalence of (c) and (a) in the Theorem, 1962). Open/open dense—open completeness is equivalent to supercompleteness. Proposition 2.5 (Burdick [3], equivalence of (c) and (a) in Theorem 2, 1991). Open/open dense—somewhere dense completeness is equivalent to supercom- pleteness. In the following table we extend this type of characterization to all 36 of the completeness properties just defined. The nine Cauchy conditions correspond to the nine rows and the four convergence conditions correspond to the four columns. We find that these 36 different combinations resolve into just five well known versions of completeness. The 36 Completeness Properties C = Complete SC = Supercomplete CC = Cofinally Complete PF = Paracompact and Fine I = Indiscrete Somewhere Open Dense Open Dense Dense Limit Limit Limit Limit Open Dense C C C CCauchy Open/ Open Dense SC SC I I Cauchy Dense/ Open Dense SC I SC I Cauchy Open CC PF I ICauchy Dense CC I PF ICauchy 102 Bruce S. Burdick The 36 Completeness Properties (continued) Somewhere Open Dense Open Dense Dense Limit Limit Limit Limit Somewhere Dense/ SC I I IOpen Dense Cauchy Somewhere Dense/Open CC I I I Cauchy Somewhere Dense/Dense CC I I I Cauchy Somewhere Dense CC I I I Cauchy We will only prove some of the results contained in the table above—enough so that the key ideas are demonstrated. First of all it is clear that all these properties imply completeness. Like- wise, dense—somewhere dense completeness clearly implies cofinal complete- ness. Going the other way we have the following two results. Proposition 2.6. Completeness implies open dense—open dense complete- ness. Proof. Suppose that (X,U) is complete and S : P → X is an open dense Cauchy ponet. Let Q be the set of open dense subsets of P and let Q = {(D,x) | x ∈ D ∈ Q}. Define a preorder on Q by (D1,x) ≤ (D2,y) if D2 ⊆ D1. Then Q is predirected. Define F : Q→ P by F(D,x) = x. Then S◦F : Q→ X is a Cauchy net. Let x be a limit of S ◦F. Then for every neighborhood O of x, S−1[O] contains an open dense set. So x is an open dense limit of S. � Proposition 2.7. A cofinally complete space is somewhere dense—somewhere dense complete. Proof. Suppose that (X,U) is cofinally complete and S : P → X is a somewhere dense Cauchy ponet. Let Q be the set of open dense subsets of P and let Q = {(D,x) | x ∈ D ∈ Q}. Define a preorder on Q by (D1,x) ≤ (D2,y) if D2 ⊆ D1. Then Q is predirected. Define F : Q → P by F(D,x) = x. Then S ◦ F : Q → X is a cofinally Cauchy net. Let x be a cluster point of S ◦ F. Then for every neighborhood O of x, S−1[O] intersects every open dense subset of P so it must be somewhere dense. So x is a somewhere dense limit of S. � We note that there is a symmetry between open and dense in the table. Here is one example of this. Partial orders in uniform spaces 103 Proposition 2.8. A space is open—open complete if and only if it is dense— dense complete. Proof. Suppose (X,U) is open—open complete. Suppose S : P → X is dense Cauchy. Let Q be the collection of dense subsets of P and let Q = {(D,x) | x ∈ D ∈ Q}. Define (D1,x) ≤ (D2,y) if D2 ⊆ D1. Define F : Q → P by F(D,x) = x. Then S ◦ F is open Cauchy. Let x be an open limit of S ◦ F. Then for every neighborhood O of x, S−1[O] is dense. So x is a dense limit of S. Suppose (X,U) is dense—dense complete. Suppose S : P → X is open Cauchy. Let Q be the collection of dense subsets of P and let Q = {(D,x) | x ∈ D ∈ Q}. Define (D1,x) ≤ (D2,y) if D2 ⊆ D1. Define F : Q → P by F(D,x) = x. Then S ◦ F is dense Cauchy. Let x be a dense limit of S ◦ F. Then for every neighborhood O of x, S−1[O] intersects every member of Q so it contain as open set. So x is an open limit of S. � Sometimes the method of the last proof doesn’t do the whole job, but the symmetry still holds. Proposition 2.9. For a uniform space the following are equivalent: (1) Supercompleteness. (2) Somewhere dense/open dense—somewhere dense completeness. (3) Dense/open dense—somewhere dense completeness. (4) Open/open dense—somewhere dense completeness. (5) Dense/open dense—dense completeness. (6) Open/open dense—open completeness. Proof. (1) implies (2). Suppose (X,U) is supercomplete. Suppose S : P → X is somewhere dense/open dense Cauchy. Let Q be the set of open dense subsets of P and let Q = {(D,x) | x ∈ D ∈ Q}. Define (D1,x) ≤ (D2,y) if D2 ⊆ D1. Then Q is predirected. Define F : Q → P by F(D,x) = x. Then S ◦ F is an almost Cauchy net. Let x be a cluster point of S ◦ F. Then for every neighborhood O of x, S−1[O] intersects every member of Q so it must be somewhere dense. So x is a somewhere dense limit of S. (2) implies (3). Trivial. (3) implies (5). A somewhere dense limit of a dense/open dense ponet will always be a dense limit. (2) implies (4). Trivial. (4) implies (6). A somewhere dense limit of an open/open dense ponet will always be an open limit. (5) implies (6). Suppose (X,U) is dense/open dense—dense complete. Sup- pose S : P → X is open/open dense Cauchy. Let Q be the collection of dense subsets of P and let Q = {(D,x) | x ∈ D ∈ Q}. Define (D1,x) ≤ (D2,y) if D2 ⊆ D1. Define F : Q→ P by F(D,x) = x. Then S ◦F is dense/open dense Cauchy. Let x be a dense limit of S ◦ F. Then for every neighborhood O of 104 Bruce S. Burdick x, S−1[O] intersects every member of Q so it contain an open set. So x is an open limit of S. (6) implies (1). This follows from Proposition 2.4. � The method of proof in Proposition 2.8 was used here to prove that (5) implies (6), but it doesn’t supply a direct proof that (6) implies (5). If we tried that, at a crucial point we would not be able to say that S ◦ F is open/open dense Cauchy. There is an asymmetry here stemming from the fact that while the F−1 image of an open set is dense, the F−1 image of a dense set merely contains an open set. In Section 3 we will see a breakdown in the symmetry of the results for this very reason (compare Corollary 3.17 with Example 3.15). The combination of paracompact and fine is equivalent to saying that every open cover is a uniform cover. This is stronger than cofinal completeness ([5], (a) implies (c) in Thereom 1). Proposition 2.10. Open—open completeness is equivalent to paracompact and fine. Proof. Suppose that (X,U) is open—open complete and that C is an open cover of X which is not uniform. Let P be the collection of all sets A ⊆ X such that no element of C contains A as a subset. Let P = {(A,x) | x ∈ A ∈ P}. Define a preorder on P by saying that (A,x) ≤ (B,y) if B ⊆ A. Define a ponet S : P → X by S(A,x) = x. P is an open Cauchy ponet. So let x be an open limit. Some O ∈C contains x. But then some A ∈ P would have to be a subset of O, a contradiction. Conversely, suppose that (X,U) is paracompact and fine. Let S : P → X be an open Cauchy ponet with no open limit. Then each point in X would have an open neighborhood O such that S−1[O] would not contain an open set. The collection of these O’s is an open cover, therefore a uniform cover. But this contradicts the open Cauchy property. � Proposition 2.11. Open/open dense—dense completeness implies that the uniform space has the indiscrete uniformity. Proof. Suppose that (X,U) is open/open dense—dense complete. Let X be given the trivial order defined by x ≤ y if and only if x = y. Then the identity map ι : X → X is an open/open dense Cauchy ponet. Let x be a dense limit. Then every neighborhood of x must contain all of X, making U the indiscrete uniformity. � 3. The functors ν,λ, and θ Now we turn to a consideration of certain functors which are associated with completeness properties. The functors ν and λ are due to Howes [9] and Ginsberg and Isbell [8], respectively. The latter functor has been utilized by many authors over the last four decades. Our functor θ was announced in the Partial orders in uniform spaces 105 book by Howes [11] without any details. This is the first time we have used it in a paper. After defining Howes’s functor ν, we give several results which illustrate the importance of ν. After that we proceed to a new definition of ν. Definition 3.1. For each infinite cardinal κ we say a uniform space (X,U) is κ-bounded if for every U ∈ U there is a subset S of X, of cardinality less than κ, with U[S] = X. For each uniform space (X,U) and each infinite cardinal κ let Uκ be the supremum of all the κ-bounded uniformites on X which are coarser than U. For an infinite cardinal κ, a topological space (X,T ) is κ-pseudocompact if every normal cover of X has a subcover of cardinality less than κ. A topological space is [κ,∞)-compact if every open cover has a subcover of cardinality less than κ. The next two definitions and several results following them are due to N. Howes. Definition 3.2. [9] Given a space (X,U) a new space (X,νU) is constructed in the following manner: embed (X,U) in its completion (Y,V), let V∗ be the finest uniformity on Y generating the same topology as V, and then let νU be the restriction of V∗ to X. It is easy to see that this construction defines a functor (in view of the definitions ν(X,U) = (X,νU) and νf = f) from the category of uniform spaces to itself. Definition 3.3. A space is preparacompact [9] if every cofinally Cauchy net has a Cauchy subnet, and it is almost preparacompact [10] if every almost Cauchy net has a Cauchy subnet. Lemma 3.4 (Howes). A space is preparacompact iff its completion is cofinally complete [9]; it is almost preparacompact iff its completion is supercomplete [10]. In [9] and [10], Howes used the functor ν to answer a question of Tamano as to which spaces had paracompact completions. Proposition 3.5 (Howes). For a space (X,U) the following are equivalent: (1) (X,U) has a paracompact completion. (2) (X,νU) is preparacompact. (3) (X,νU) is almost preparacompact. This in turn allowed him to characterize the spaces with Lindelöf completions in several different ways. Proposition 3.6 (Howes). For a space (X,U) the following are equivalent: (1) (X,U) has a Lindelöf completion. (2) (X,νU) is preparacompact and ℵ1-bounded. (3) (X,νU) is almost preparacompact and ℵ1-bounded. 106 Bruce S. Burdick The following generalizes these results of Howes’s. Lemma 3.7. For a space (X,U) and an infinite cardinal κ, the following are equivalent: (1) (X,U) has a κ-pseudocompact completion. (2) (X,νU) is κ-bounded. Proposition 3.8. For a space (X,U) and an infinite cardinal κ, the following are equivalent: (1) (X,U) has a paracompact, [κ,∞)-compact completion. (2) (X,νU) is κ-bounded and preparacompact. (3) (X,νU) is κ-bounded and almost preparacompact. These results are mentioned to demonstrate the usefulness of the functor ν. Therefore we should ask if there are other ways of defining ν, ways which might facilitate the use of the characterizations above. We give new constructions of ν which involve other uniformities on the given set X but do not require adding any more points to X. We feel that the results above, which give properties of the completion of a space, will be more significant if the construction of ν itself does not make use of the completion. Proposition 3.9. For a fixed uniform space (X,U) and a possibly different uniformity U∗ on X, the following are equivalent: (1) Any ponet which is open dense Cauchy for U is open dense Cauchy for U∗. (2) Any net which is Cauchy for U is Cauchy for U∗. (3) Any filter which is Cauchy for U is Cauchy for U∗. (4) U∗ ⊆ νU. Proof. (1) implies (2). Trivial. (2) implies (3). Elementary. (3) implies (4). Let the Hausdorff completion of (X,U) be i : (X,U) → (Y,V) and the Hausdorff completion of (X,U∗) be i∗ : (X,U∗) → (Z,W). It suffices to show that there is a continuous map f : (Y,T (V)) → (Z,T (W)) where f ◦ i = i∗. The sets Y and Z may be regarded as sets of equivalence classes of Cauchy filters, and the equivalence relation is such that two filters F and F′ are equivalent if and only if F∩F′ is Cauchy. So we may define f by saying that if F is a representative of the equivalence class [F]U ∈ Y , then f([F]U) = [F]U∗. Our remarks above show that f is well-defined. To show f continuous it suffices to show that if filter F converges to y in (Y,V) then f[F] converges to f(y) in (Z,W). This is certainly true if every F ∈F intersects i[X], since the trace of F on i[X] would be U-Cauchy, therefore i−1[F] would be a representative of the equivalence class y, and so also of f(y), and so F = i[i−1[F]] would converge to f(y). So given F converging to y in (Y,V), let G = {V [F] | V ∈ V,F ∈ F}. G still converges to y, so it is a Cauchy filter on (Y,V) and every member of G Partial orders in uniform spaces 107 intersects i[X]. So i−1[G] is Cauchy on (X,U). Since i−1[G] is a representative of the equivalence class y it is also a representative of the equivalence class f(y). Then for any W ∈ W, W [y] will contain G ∩ i[X] for some G ∈ G. Suppose G = V [F] for some F ∈F. Then every point of F is the limit with respect to (Y,V) of a filter on G∩ i[X] and so by the remark in the last paragraph every point of f[F] is the limit with respect to (Z,W) of a filter on f[G]. If W has been chosen to be a closed relation then W [f(y)] contains f[F]. This shows that f[F] converges to f(y). (4) implies (1). Any ponet which is open dense Cauchy for U will have an open dense limit in the completion (X,U) of (X,U), by the table in Section 2. This will still be an open dense limit when U is replaced by the fine uniformity for its topology. Therefore the ponet will be open dense Cauchy for νU and so for U∗. � Corollary 3.10. For any uniform space (X,U), νU is equal to (1) the supre- mum of the uniformities U∗ such that every U-Cauchy net is U∗-Cauchy, and (2) the supremum of the uniformities U∗ such that every U-open dense Cauchy ponet is U∗-open dense Cauchy. In each case the supremum is the finest mem- ber of the set. Corollary 3.11. A uniform space (X,U) has a paracompact completion if and only if every cofinally Cauchy net in (X,νU) has a subnet which is Cauchy for U. Observation 3.12. If a space (X,U) is complete then νU is fine; if the space is paracompact (or even just topologically complete) then the converse is true. Another useful functor is the locally fine coreflection, λ. The reader is re- ferred to [8], [11], [12], [13], and [15] for many properties of this functor. We will characterize λ using ponets as we have for ν. Definition 3.13. A cover C of X is called uniformly locally uniform if there is a uniform cover C′ such that on each S ∈C′, the trace of C on S is uniform. A space (X,U) is called locally fine if every uniformly locally uniform cover is uniform. Given a space (X,U) the uniformity λU is the coarsest one finer than U such that (X,λU) is locally fine. λ is constructed by transfinite recursion (see [8] or [13] ). Proposition 3.14. νλ = ν. Consequently, for a uniform space (X,U), we have νU finer than λU. Proof. It suffices to show that the completion of λ(X,U) is homeomorphic to the completion of (X,U) via a homeomorphism which is the identity on X. This follows from Ginsberg’s and Isbell’s ([8], Theorem 4.4) (λ commutes with completion). � We wish to prove some characterizations of λ similar to those we have done for ν, but first we observe an example that shows that the obvious net property doesn’t hold in this case. 108 Bruce S. Burdick Example 3.15. Stable filters for (X,U) need not be stable for λU, and conse- quently almost Cauchy nets for (X,U) need not be almost Cauchy for λU, nor do dense/open dense ponets for (X,U) need to be dense/open dense for λU. Let U be the usual uniformity on the reals, and let a filter F be generated by the sets F� = ⋃ n∈Z[n − �,n + �], for � > 0. Then F is U-stable. Since U is a complete metric uniformity, λU is fine [8]. But even if U∗ is a uniformity which makes the function f : (R,U∗) → (R,U), where f(x) = x2, uniformly continuous, then F is not U∗-stable. So F is not λU-stable. Proposition 3.16. For a fixed uniform space (X,U) and a possibly different uniformity U∗ on X, the following are equivalent: (1) U∗ ⊆ λU. (2) Any ponet which is open/open dense Cauchy for U is open/open dense Cauchy for U∗. (3) Any ponet which is open/open dense Cauchy for U is open Cauchy for U∗. (4) Any locally fine uniformity which is finer than U is finer than U∗. Proof. (1) implies (2). Given a ponet which is open/open dense Cauchy for U we can prove by transfinite induction that it is open/open dense Cauchy for λU. This is the essence of Lemma 40 of Chapter VII of [13]. (2) implies (3). Trivial. (3) implies (4). Given a locally fine uniformity V on X with U ⊆V, suppose that U∗ 6⊆ V. Then there is a U∗-uniform cover C which is not V-uniform. Let P be the collection of all sets A ⊆ X such that C is not a V-uniform cover of A. Let P = {(A,x) | x ∈ A ∈ P}. Define a preorder on P by saying that (A,x) ≤ (B,y) if B ⊆ A. Define a ponet S : P → X by S(A,x) = x. We show that S is open/open dense Cauchy for V. It suffices to show that for any A ∈ P and U ∈V there is a U-small B ∈ P with B ⊆ A. Suppose not, i.e., every U-small subset of A is V-uniformly covered by C. Consider the cover C′ = {C ∪ (X − A) | C ∈ C}. The U-small subsets of X are all V-uniformly covered by C′, so by local fineness C′ is a V-uniform cover of X. The trace of C′ on A is the same as the trace of C, so C is a V-uniform cover of A, a contradiction. Since S is V-open/open dense Cauchy it is U-open/open dense Cauchy. But it fails to be U∗-open Cauchy since no element of P is contained in any element of C, and this contradicts property (3). (4) implies (1). This follows since λU is a locally fine uniformity which is finer than U. � Corollary 3.17. For any uniform space (X,U), λU is equal to (1) the supre- mum of the uniformities U∗ such that every U-open/open dense Cauchy ponet is U∗-open/open dense Cauchy, and (2) the supremum of the uniformities U∗ such that every U-open/open dense Cauchy ponet is U∗-open Cauchy. In each case the supremum is the finest member of the set. Partial orders in uniform spaces 109 The usual definition of λ gives rise to technical difficulties in that the Gins- berg-Isbell derivatives used in the transfinite induction need not be covering uniformities, and then it is tricky to show in the end that their union, i.e., the λ uniform covers, is a covering uniformity. If property (1) in Corollary 3.17 were taken as the definition of λU instead this wouldn’t be a problem. What’s more it would be easy to show directly that λU preserves the open/open dense ponets of U. The next theorem of Isbell’s [12] is our chief reason for interest in the functor λ. Isbell’s Theorem A space (X,U) is supercomplete iff it is paracompact and λU is fine. The fact that paracompactness plus fineness implies supercompleteness, and that supercomplete implies paracompact, can be proved without use of the functor λ. If we add to these facts the observation that λ preserves the topology of the space (true because it is true for ν) then Isbell’s Theorem follows from our Corollary 3.17 and Propostion 2.9. We should not claim too much, however. We have not replaced the traditional development of this subject because we have used previously known properties of λ in the proofs of Propositions 3.14 and 3.16. In particular we used the assumption that λ as traditionally defined is in fact a uniformity to prove that (4) implies (1) in Proposition 3.16. Let us further point out that Pelant in [15] has shown that the locally fine uniform spaces are all subfine and so λ coincides with the subfine coreflection described in Chapter VII of [13]. This also then provides a relatively straight- forward way of defining λ. However, the proof that it coincides with the original definition of λ is quite involved. Our Corollary 3.17 has the combined advan- tages of being relatively easy to prove and giving a construction of λU which clearly yields a uniformity. In view of these properties of ν and λ it would be interesting to have a third functor θ with the following properties: (1) Cofinal completeness should be equivalent to paracompactness and θ fine. (2) θ should be coarser than λ. (3) θ2 = θ. (4) θ shouldn’t change the underlying set or the generated topology, and should make the uniformity finer than before. (5) θ, like ν and λ, should be a coreflection when considered as a functor from the category of all uniform spaces to its image category. (This actually follows from (3) and (4) and the assumption that θ is a func- tor). Definition 3.18. A cover C of a uniform space (X,U) is uniformly locally finitizible if there is a U ∈ U such that for any x ∈ X, U[x] is covered by some finite subset of C. A uniform space (X,U) is ℵ0-nearly discrete if for any uniformity U∗ on X, if every U∗-uniform cover is U-uniformly locally finitizible then U∗ ⊆ U. For a uniform space (X,U) let θ+U be the supremum of the 110 Bruce S. Burdick uniformities U∗ on X such that every U∗-uniform cover C is U-uniformly locally finitizible. Lemma 3.19. For a uniform space (X,U), θ+U is ℵ0-nearly discrete and any θ+U-uniform cover is U-uniformly locally finitizible. Proposition 3.20. For a fixed uniform space (X,U) and a possibly different uniformity U∗ on X, the following are equivalent: (1) U∗ ⊆ θ+U. (2) Every cofinally Cauchy net for U is cofinally Cauchy for U∗. (3) Every weakly Cauchy filter for U is weakly Cauchy for U∗. (4) Every somewhere dense Cauchy ponet for U is somewhere dense Cauchy for U∗. (5) Every open Cauchy ponet for U is somewhere dense Cauchy for U∗. (6) Every dense Cauchy ponet for U is somewhere dense Cauchy for U∗. (7) Every ℵ0-nearly discrete uniformity which is finer than U is finer than U∗. Proof. (1) implies (2). Given a U-cofinally Cauchy net S : D → X and a U∗- uniform cover C, take a U-uniform cover C′ such that each member of C′ can be covered by finitely many members of C. There is some C ∈ C′ such that S is frequently in C, and then among the finitely many members of C that cover C, S must be frequently in at least one of them. Equivalence of (2) and (3). Elementary. (2) implies (4). This uses the methods of Section 2. (4) implies (5). Trivial. Equivalence of (5) and (6). This uses the methods of Section 2. (5) implies (7). Let V be an ℵ0-nearly discrete uniformity on X with U ⊆V. Suppose that U∗ 6⊆ V. Then there is a U∗-uniform cover C which is not V- uniform, so it is not V-uniformly locally finitizible. Let P be the collection of all sets A ⊆ X such that C has no finite subset covering A. Let P = {(A,x) | x ∈ A ∈ P}. Define a preorder on P by saying that (A,x) ≤ (B,y) if B ⊆ A. Define a ponet S : P → X by S(A,x) = x. S is clearly V-open Cauchy so it is U-open Cauchy. To show it is not U∗- somewhere dense Cauchy we observe that for any A ∈ P and any C ∈ C, A−C ∈ P. (7) implies (1). This follows from Lemma 3.19. � Corollary 3.21. A uniform space (X,U) which satisfies νU ⊆ θ+U has a paracompact completion if and only if it is preparacompact. We should point out that unlike ν and λ above, θ+ may change the topology of the space since it always contains all the totally bounded uniformities for the discrete topology on the given set of points. Definition 3.22. For any uniform space (X,U) define θU = θ+U ∧λU. Partial orders in uniform spaces 111 Observation 3.23. θ2 = θ since the same is true for θ+ and λ. θ preserves topology because the same is true for λ. The following proposition is an immediate consequence of Propositions 3.16 and 3.20. Proposition 3.24. For a fixed uniform space (X,U) and a possibly different uniformity U∗ on X, the following are equivalent: (1) U∗ ⊆ θU. (2) Every cofinally Cauchy net for U is cofinally Cauchy for U∗ and ev- ery ponet which is open/open dense Cauchy for U is open/open dense Cauchy for U∗. (3) Every somewhere dense Cauchy ponet for U is somewhere dense Cauchy for U∗ and every ponet which is open/open dense Cauchy for U is open/open dense Cauchy for U∗. Corollary 3.25. For any uniform space (X,U), θU is equal to (1) the supre- mum of the uniformities U∗ such that every U-open/open dense Cauchy ponet is U∗-open/open dense Cauchy and every cofinally Cauchy net for U is cofi- nally Cauchy for U∗, and (2) the supremum of the uniformities U∗ such that every U-open/open dense Cauchy ponet is U∗-open/open dense Cauchy and ev- ery somewhere dense Cauchy ponet for U is somewhere dense Cauchy for U∗. In each case the supremum is the finest member of the set. Definition 3.26. We will say (see Rice’s paper [16] ) a space is uniformly paracompact if every open cover has a uniformly locally finite refinement. Proposition 3.27. For a uniform space (X,U) the following are equivalent: (1) (X,U) is cofinally complete. (2) Any open cover O of X which is closed under finite unions is U- uniform. (3) X is paracompact and all locally finite collections in X are U-uniformly locally finite. (4) (X,U) is uniformly paracompact. Proposition 3.27 is a combination of several results in Fried [6]. See our paper [3] for more details about the history of this result. In [16], Rice gave several different ways of characterizing uniform paracom- pactness. Some of his results are suggestive of Isbell’s Theorem, but he stops short of defining a functor θ analogous to λ. We will prove a new version of Rice’s Theorem 3 which uses only functors into the category of uniformities. Definition 3.28. Let κ be an infinite cardinal. A space (X,U) is locally κ-fine if every U-uniformly locally Uκ-uniform cover is a U-uniform cover. Froĺık [7] has already called this locally p-fine when κ = ℵ0 and locally e-fine when κ = ℵ1. He states without proof that these properties are coreflective. Note that among the uniformities U∗ such that every U∗-uniform cover is a U-uniformly locally Uκ-uniform cover, there is a finest one. 112 Bruce S. Burdick Definition 3.29. For a uniform space (X,U) let θ−U be the supremum of all the uniformities U∗ whose uniform covers are all U-uniformly locally Uℵ0 - uniform. θ− is a functor but it doesn’t always satisfy θ2− = θ−. Example 3.30. A space where θ2−U 6= θ−U. Let X = ω × ω × 2. Let the uniformity U be generated by all equivalences relations satisfying the following properties: (1) For all but finitely many pairs (n,m), (n,m, 0) is related to (n,m, 1). (2) For all but finitely many n, (n,m,i) is related to any (n,m′, i′). The relation R which relates (n,m,i) to (n′,m′, i′) if and only if i = i′ is not a member of θ−U. But the relations Rk which have as their equivalence classes the three sets {(n,m,i) | n ≤ k, i = 0}, {(n,m,i) | n ≤ k, i = 1}, and {(n,m,i) | n > k}, are to be found in θ−U, and it is because of them that R is a member of θ2−U. Lemma 3.31. For any uniformity U we have θ−U ⊆ θU. Proposition 3.32. For any uniform space (X,U) the following are equivalent: (1) (X,U) is cofinally complete. (2) (X,θ−U) is paracompact and fine. (3) (X,θU) is paracompact and fine. Proof. (1) implies (2). Suppose (X,U) is cofinally complete. Given a cover C of X, let C′ be the collection of open sets O such that there is a U ∈U where for any x ∈ X, U[x] is a subset of some member of C. C′ is a cover of X and it is closed under finite unions. So by Proposition 3.27 it is uniform. This shows that any open cover of X is uniformly locally uniform. Again by Proposition 3.27, every open cover of X has a uniformly locally finite refinement. So every open cover of X has a refinement which is U- uniformly locally Uℵ0 -uniform. Therefore the fine uniformity is one of the uniformities which we take the supremum of to get θ−U. Since θ− doesn’t change the topology, θ−U must be the fine uniformity. The fact that cofinal completeness implies paracompactness completes the proof. (2) implies (3). θ is trapped between θ− and λ. Therefore if θ− is fine so must θ be fine as well. (3) implies (1). If a net is cofinally Cauchy for U it is cofinally Cauchy for θU by Proposition 3.24. Then since paracompact and fine implies (X,θU) is cofinally Cauchy, the net must have a cluster point. � Proposition 3.33. For a fixed uniform space (X,U) and a possibly different uniformity U∗ on X, the following are equivalent: (1) Any open Cauchy ponet for U is open Cauchy for U∗. (2) Any dense Cauchy ponet for U is dense Cauchy for U∗. Partial orders in uniform spaces 113 (3) U∗ ⊆U. Proof. (1) implies (2). Suppose that U∗ 6⊆ U. Then there is a cover C which is U∗-uniform but not U-uniform. Let P be the set of A ⊆ X such that no C ∈C has A as a subset. Proceeding as in several other proofs in this section we can construct a ponet which is U-open Cauchy but not U∗-open Cauchy. (1) implies (2). Use the methods of Section 2. (3) implies (1). Trivial. � Omnibus Theorem A uniform space is complete (supercomplete, cofinally complete, paracompact and fine) if and only if the supremum of the uniformi- ties which preserve the open dense (the open/open dense, both the open/open dense and the somewhere dense, the open) Cauchy ponets is fine and the un- derlying topological space is topologically complete (paracompact, paracompact, paracompact ). If this supremum is complete (supercomplete, cofinally complete, paracompact and fine) then so is the original space. References [1] P. Alexandroff, Diskrete Räume, Matematicheskij Sbornik 2-44 (1937), 501–519. [2] Bruce S. Burdick, A note on completeness of hyperspaces, in General Topology and Applications, Fifth Northeast Conference, S. J. Andima, et al. (eds.), Lecture Notes in Pure and Applied Mathematics 134 (Marcel Dekker, New York, 1991), 19–24. [3] Bruce S. Burdick, On linear cofinal completeness, Topology Proceedings 25 (2000), 435–455. [4] Á. Császár, Strongly complete, supercomplete and ultracomplete spaces, in Mathematical Structures—Computational Mathematics—Mathematical Modelling, Papers dedicated to Professor L. Iliev’s 60th Anniversary, Sofia, 1975, 195–202. [5] H. Corson, The determination of paracompactness by uniformities, American Journal of Mathematics 80 (1958), 185–190. [6] Jan Fried, On paracompactness in uniform spaces, Commentationes Mathematicae Uni- versitatis Carolinae 26 (1985), 373–385. [7] Zdenĕk Froĺık, Locally e-fine measurable spaces, Trans. Amer. Math. Soc. 196 (1974), 237–247. [8] Seymour Ginsberg and J. R. Isbell, Some operators on uniform spaces, Trans. Amer. Math. Soc. 93 (1959), 145–168. [9] N. Howes, On completeness, Pacific J. Math. 38 (1971), 431–440. [10] N. Howes, Paracompactifications, preparacompactness, and some problems of K. Morita and H. Tamano, Questions and Answers in General Topology 10 (1992), pp. 191–204. [11] N. Howes, Modern Analysis and Topology, (Springer-Verlag, New York, 1995). [12] J. R. Isbell, Supercomplete spaces, Pacific J. Math. 12 (1962), 287–290. [13] J. R. Isbell, Uniform Spaces, (Amer. Math. Soc., Providence, 1964). [14] John L. Kelley, General Topology, (Van Nostrand, Princeton, 1955). [15] Jan Pelant, Locally fine uniformities and normal covers, Czechoslovak Math. J. 37 (112) (1987), 181–187. [16] Michael D. Rice, A note on uniform paracompactness, Proc. Amer. Math. Soc. 62 (1977), 359–362. [17] Mary Ellen Rudin, Lectures on Set Theoretic Topology, (Amer. Math. Soc., Providence, 1975). 114 Bruce S. Burdick Received February 2002 Revised February 2003 Bruce S. Burdick Department of Mathematics, Roger Williams University, Bristol, Rhode Island 02809, USA. E-mail address : bburdick@rwu.edu On the use of partial orders in uniform spaces. By Bruce S. Burdick