p-REGULAR CAUCHY COMPLETIONS

This paper gives a further development of p-regular completion theory, including a study ofp-regular Reed completions, the role of diagonal axioms, and the relationship between p-regular and p-topological completions.


Introduction. Since a paper of p-regular completions
first appeared in 1991, some recent and relevant developments have occurred which provide the motivation for this paper.The discovery of a dual relationship between the properties "pregular" and "p-topological" in convergence space theory led to an investigation of p-topological convergence spaces [11] and p-topological Cauchy completions [10].These, in turn, raised some questions about p-regular completion theory that had not been previously considered, such as the possible duality between p-regular and p-topological Cauchy spaces, the role of diagonal axioms in the study of p-regular completions, the existence of p-topological Reed completions, and the relevance of p-regular completions to the study of regular completions.These topics, along with a comparison of the behavior of p-regular and p-topological completions, form the subject matter of this paper.

The fine p-regular completion.
For background information on convergence or Cauchy spaces, the reader is referred to [1] or [5].Additional information about p-regular and p-topological Cauchy and convergence spaces may be found in [4,10,11].If (X, Ꮿ) is a Cauchy space, the associated convergence structure is denoted by q Ꮿ .We shall assume that all convergence spaces, Cauchy spaces, and Cauchy completions discussed in this paper are T 2 (i.e., Hausdorff) unless otherwise indicated.
If X is any set, then F(X) denotes the set of all filters on X, and if x ∈ X, then ẋ denotes the ultrafilter generated by {x}.Let C(X) be the set of all convergence structures on X.
Let (X, Ꮿ) be a Cauchy space, X * the set of all Cauchy equivalence classes, and X = {[Ᏺ] ∈ X * : Ᏺ is not q Ꮿ -convergent}.The canonical map j : X → X * , given by j(x) = [ ẋ], is an injection.It is proved in [8] that every completion of (X, Ꮿ) is equivalent to one in standard form, which means that X * is the underlying set for the completion, j the Cauchy embedding map, and j(Ᏺ) ∩ [ Ᏺ] is a Cauchy filter in the completion structure for every Ᏺ ∈ Ꮿ.Consequently, we shall limit our attention in this paper to completions in standard form.
If (X, Ꮿ) is a Cauchy space and p ∈ C(X), then (X, Ꮿ) is defined to be p-regular if Ᏺ ∈ Ꮿ implies cl p Ᏺ ∈ Ꮿ, and p-topological if Ᏺ ∈ Ꮿ implies there is Ᏻ ∈ Ꮿ such that Ᏺ ≥ I p Ᏻ (where cl p and I p are the closure and interior operators for p).(X, Ꮿ) is regular (respectively, topological) if it is q Ꮿ -regular (respectively, q Ꮿ -topological).The convergence structure p is extended to a convergence structure p * on X * by defining p * to be the finest convergence structure on X * satisfying: (1) j(Ᏺ) A completion of (X, Ꮿ) is p-regular (respectively, p-topological) if the completion space (X * , Ᏸ) is p * -regular (respectively, p * -topological).Note that a q Ꮿ -regular (or q Ꮿ -topological) completion does not have the same meaning as a regular (or topological) completion.
We next introduce some additional notation and terminology relevant to the study of p-regular completions.Given a Cauchy space (X, Ꮿ), p ∈ C(X), and A ⊆ X, we define A Cauchy space (X, Ꮿ) is said to have property S if, whenever Ᏺ, Ᏻ ∈ Ꮿ and [Ᏺ] ≠ [Ᏻ], θ Ꮿ Ᏺ ∨ θ Ꮿ Ᏻ fails to exist.Note that θ Ꮿ Ᏺ ∨ θ Ꮿ Ᏻ fails to exist if either of these filters fails to exist or if both exist but contain mutually disjoint sets.Property S is a separation property stronger than T 2 (since (X, Ꮿ) is additionally assumed to be T 2 ), but weaker than Cauchy separated, which requires the existence of a Cauchycontinuous, real-valued function which separates the non-equivalent Cauchy filters.
Given a Cauchy space (X, Ꮿ) and p ∈ C(X), let Ꮿ * p be the Cauchy structure on X * generated by {∆ p Ᏺ : Ᏺ ∈ Ꮿ}.Although Ꮿ * p is generally not T 2 we have the following result.
From the given assumptions about (X, Ꮿ), we can deduce that ∆ p Ᏼ 1 ,...,∆ p Ᏼ n being linked implies that and so there is To prove the last statement, note that j : (X, Ꮿ) → X * , Ꮿ * p is obviously a Cauchycontinuous bijection, and Cauchy-continuity of j −1 follows by Lemma 2.1.Also, j(X) is obviously dense in X * .
Recall that a completion (X * , Ᏸ) of a Cauchy space (X, Ꮿ) is strict if, for each Ᏼ ∈ Ᏸ, there is Ᏺ ∈ Ꮿ such that cl qᏰ jᏲ ≤ Ᏼ. Proposition 2.3.If (X, Ꮿ) is a p-regular Cauchy space satisfying S, then X * , Ꮿ * p is the finest p-regular completion of (X, Ꮿ).This completion is strict, and is in the fact the only strict p-regular completion (X, Ꮿ).
Proof.If Ᏺ ∈ Ꮿ, it is clear that ∆ p Ᏺ must belong to every p-regular completion of (X, Ꮿ), which proves the first assertion.The second statement is proved in [5, Propositions 2.6 and 2.7].
In view of Proposition 2.3, we shall call X * , Ꮿ * p the fine p-regular completion of a p-regular Cauchy space (X, Ꮿ) with property S.

p-regular Reed completions.
In 1971, Reed [9], defined a family of completions of a Cauchy space (X, Ꮿ), which we shall now describe.By a Reed selection function, we shall mean a function λ : X * → F(X) which satisfies the following conditions: Let Λ be the set of all Reed selection functions on (X, Ꮿ), and for arbitrary λ ∈ Λ, F ⊆ X, and Ᏺ ∈ F(X), let F λ = {z ∈ X * : F ∈ λ(z)}, and let Ᏺ λ be the filter on X * generated by {F λ : F ∈ Ᏺ}.For λ ∈ Λ and Γ ⊆ Λ, the following complete Cauchy structures are defined on X * : [9] that for any Γ ⊆ Λ, (X * , Ꮿ Γ ) is a completion of (X, Ꮿ) with an extension property relative to Cauchy-continuous maps into complete, regular Cauchy spaces.The set X * , Ꮿ Γ : Γ ⊆ Λ} (including those of the form (X * , Ꮿ λ ), where Γ = {λ}, λ ∈ Λ) is called the Reed family of completions for (X, Ꮿ).By their construction, it is clear that all Reed completions are strict.
In 1984, Colebunders [7], studied conditions under which Kowalsky completion and other Reed completions are regular.We shall apply her approach to the study of pregular Reed completions.Given a Cauchy space (X, Ꮿ), p ∈ C(X), λ ∈ Σ, and F, A ⊆ X, we proceed to define some relevant notation and terminology.[9] that for a relatively round space, Ꮿ Γ = Ꮿ Σ , for all Γ ⊆ Σ. Proposition 3.4.Let (X, Ꮿ) be a Cauchy space, p ∈ C(X), λ ∈ Σ, and Ᏺ ∈ F(X), then (a) Proof of (b).Let F and A be subsets of X; it suffices to show F < λ A if and only if F < λ q Ꮿ A. Assuming F < λ A and x ∈ cl q Ꮿ F , there is Ᏼ and hence A ∈ λ(z).Theorem 3.5.Let (X, Ꮿ) be a p-regular Cauchy space, p ∈ C(X), and λ ∈ Σ.
Conversely, the assumed condition implies property S, so it remains to show that . The converse holds because the supremum of a set of p-regular Cauchy structures is p-regular (see [5]).It was shown in [7] that for a regular Cauchy space (X, Ꮿ) and λ ∈ Λ, X * , Ꮿ * λ is a regular completion if and only if Ᏺ ∈ Ꮿ implies s λ Ᏺ ∈ Ꮿ. Combining this result with Proposition 3.4 and Theorem 3.5, we obtain the next corollary.Corollary 3.8.If (X, Ꮿ) is regular Cauchy space and Γ ⊆ Σ, then (X * , Ꮿ Γ ) is regular if and only if (X * , Ꮿ Γ ) is q Ꮿ -regular.

Diagonal axioms.
A form of "duality" between the properties "regular" and "topological" in convergence space theory was first observed by Cook and Fischer [2] in 1967.Both properties have "diagonal" characterizations and if the concluding implication in the characterizations of either property is reserved, the resulting axiom characterizes the other property.This dual behaviour is shown in [11] to extend to the properties "p-regular" and "p-topological" in the convergence space setting.We begin this section by showing that properties are likewise dual in the setting of Cauchy spaces.
Let (X, Ꮿ) be a Cauchy space, J an arbitrary set, and let ρ : J → F(X) be a "selection function."Then κρᏲ is defined to be the filter ∪ Next, for a Cauchy space (X, Ꮿ) and p ∈ C(X), we define the following dual axioms.
∆ p : Let J be any set, ψ : J → X * , and ρ : J → F(X) be such that ρ(y ∆p : Let J be any set, ψ : J → X * , and ρ : J → F(X) be such that ρ(y) Let U(X) denote the set of all ultrafilters on X.
(1) (X, Ꮿ) is p-regular if and only if ∆p is satisfied.
(2) (X, Ꮿ) is p-topological if and only if ∆ p is satisfied.
Combining the preceding result with [10, Proposition 1.2 and Corollary 2.8], we obtain the next corollary.topological completions are strict, p-regular completions are generally not, although every Cauchy space which has a p-regular completion also has a strict one, namely X * , Ꮿ * p .We should also keep in mind that every p-topological Cauchy space must satisfy q Ꮿ ≤ p (see [10]), whereas no such restriction applies to p-regular Cauchy spaces or completions.

5.
Regular versus q Ꮿ -regular completions.We first note the obvious fact that a Cauchy space is regular (respectively, topological) if and only if it is q Ꮿ -regular (respectively, q Ꮿ -topological).However this observation does not extend to Cauchy completions.
(1) A regular completion of (X, Ꮿ) is also a q Ꮿ -regular completion, but the converse is generally false.
(2) A topological completion of (X, Ꮿ) is also q Ꮿ -topological completion, but the converse is generally false.
Proof.(1) If (X * , Ᏸ) is a regular completion of (X, Ꮿ), then Ᏼ ∈ Ᏸ implies cl qᏰ Ᏼ = cl q * Ꮿ Ᏼ ∈ Ᏸ, and cl qᏰ j(X) = cl q * Ꮿ j(X) = X * , so (X * , Ᏸ) is also a q Ꮿ -regular completion.An example is given in [3] of a Cauchy space (X, Ꮿ) with a regular completion but no strict regular completion.The existence of a regular completion implies that (X * , Ꮿ * q Ꮿ ) is a q Ꮿ -regular completion which, being strict, cannot be regular.
(2) The first part of the assertion is easily verified, and an example is found in [10] of a q Ꮿ -topological completion which is not topological.
However, for Reed completion, we obtain the following result by [10, Theorem 4.7 and Corollary 3.8].
Proposition 5.2.Let (X * , Ꮿ Γ ) be an arbitrary Reed completion of (X, Ꮿ).(a) (X * , Ꮿ Γ ) is a regular completion if and only if it is a q Ꮿ -regular completion.(b) (X * , Ꮿ Γ ) is a topological completion if and only if it is a q Ꮿ -topological completion.
The next proposition is a restatement of [7,Theorem 3.4(b)]; it is repeated here in contrast to the corresponding situation for regular completions, as shown in Example 5.4 Proposition 5.3 [10].A topological Cauchy space (X, Ꮿ) has a topological completion if and only if it has a q Ꮿ -topological completion.
In the example that follows, we construct a regular Cauchy space (X, Ꮿ) with property S, which therefore has X * , Ꮿ * q Ꮿ as a strict, q Ꮿ -regular completion.Let (X * , Ꮿ * r ) denote the regular modification of the Wyler completion of (X, Ꮿ).We shall show that (X * , Ꮿ * r ) is not T 2 , and it follows by [3, Proposition 1.2], that (X, Ꮿ) has no regular completion (with or without the T 2 property).For readers acquainted with the regularity series (see [9]) of a convergence space, it is of interest to note that (X * , Ꮿ * q Ꮿ ) is the first term of the regularity series of the Wyler completion.In the next example, it turns out that the T 2 property is "lost" in the transition between the first and the second terms of the regularity series of the Wyler completion, and consequently (X * , Ꮿ * r ) is not T 2 .
Example 5.4.Let X be an infinite set.Let {A k : k ∈ N}, {B k : k ∈ N}, {C k : k ∈ N}, and {D k : k ∈ N} be denumerable collections of pairwise disjoint, denumerable subsets of X, such that each member of each collection has an empty intersection with every member of each of the other collections.We first define some subsets of X: k is generated by {D k \D kj : j ∈ N}, for all k ∈ N. Let Ꮿ be the Cauchy structure on X generated by: { ẋ : x ∈ X}∪{Ᏺ}∪{Ᏻ}∪{Ꮽ kj ∩Ꮿ kj : k, j ∈ N} ∪ {Ꮾ kj ∩ Ᏸ kj : k, j ∈ N} ∪ {Ᏼ k ∩ k : k ∈ N}.Since q Ꮿ is obviously discrete, (X, Ꮿ) is regular.The proof that (X, Ꮿ) satisfies S is tedious and requires examining a number of cases; we omit these details.From these observations, we conclude that X * , Ꮿ * q Ꮿ is a q Ꮿ -regular completion of (X, Ꮿ).Let q * r be the convergence structure on X * derived from Ꮿ * r , the regular modification of Ꮿ w .It is clear that [Ᏺ] and [Ᏻ] are distinct elements of X * , but one can show that cl p * r jᏲ p * r -converges to both of these elements, so that (X * , Ꮿ * r ) is not T 2 .From the remarks preceding the example, it follows that (X, Ꮿ) has a regular completion.
The fact that the analogue of Proposition 5.3 for regular Cauchy spaces fails, as evidenced by Example 5.4, is perhaps related to another problem: that of finding a diagonal axiom which characterizes Cauchy spaces having a regular (or strict, regular) completion.An effort to solve this problem for strict, regular completion was recently made by Brock and Richardson, [1], but the diagonal condition that they found has to be combined with two additional conditions in order to characterize Cauchy spaces having strict, regular completions.Indeed, this problem remains unsolved even for p-regular completions, since our results in Corollary 4.2(a), like that of [1], requires combining an additional property along with ∆p in order to characterize Cauchy spaces allowing p-regular completions.

Corollary 3 . 7 .
If (X, Ꮿ) is a Cauchy space with a p-regular completion, X * , Ꮿ * p is a Reed completion if and only if there is λ ∈ Σ such that Ᏺ ∈ Ꮿ implies s p λ Ᏺ ∈ Ꮿ.
.2) Assume further that each A k is partitioned into a denumerable number of denumerable sets {A kj : j ∈ N}, and carry out the same partitions with the corresponding notation for B k , C k , and D k , for all k ∈ N.We next define the following filters on X: Ᏺ is generated by {F n : n ∈ N}; Ᏻ is generated by {G n : n ∈ N}; Ꮽ kj is generated by {A kj \F : F finite}, for all k, j ∈ N; Ꮾ kj is generated by {B kj \F : F finite}, for all k, j ∈ N; Ꮿ kj is generated by {C kj \F : F finite}, for all k, j ∈ N; Ᏸ kj is generated by {D kj \F : F finite}, for all k, j ∈ N; Ᏼ k is generated by {C k \C kj : j ∈ N}, for all k ∈ N;