p-TOPOLOGICAL CAUCHY COMPLETIONS

The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “p-regular” and “p-topological.” Since earlier papers have investigated regular,p-regular, and topological Cauchy completions, we hereby initiate a study of p-topological Cauchy completions. A p-topological Cauchy space has a p-topological completion if and only if it is “cushioned,” meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing a p-topological completion, it is shown that a certain class of Reed completions preserve the p-topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsest p-topological completions. However, not all p-topological completions are Reed completions. Several extension theorems for p-topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowing p-topological and p′topological completions, respectively, can always be extended to a θ-continuous map between any p-topological completion of the first space and any p′-topological completion of the second.

Introduction.The duality between "regular" and "topological" as convergence space properties was recognized only recently through the further study of certain "diagonal axioms" initially introduced by Kowalsky [7], Cook and Fischer [1].The axioms R and F which characterize these two properties and reveal their duality are discussed in [4] and [6].A subsequent characterization of "topological" using the notion of "neighborhood filter of a filter" (see [10]) makes this duality apparent using simpler and more familiar characterizations of these properties than those provided by the diagonal axioms.
The same dual relationship extends to generalizations of these properties which are called "p-regular" and "p-topological."The notion of a p-regularity was introduced in 1990 by Kent and Richardson [2] for reasons unrelated to the aforementioned duality.A convergence space (X, q) is p-regular relative to another convergence structure p on X if the p-closure of a filter q-converges to a point whenever the filter itself q-converges to that point.It was shown in [2] that p-regularity is significant; both because of its nice structural properties and also the fact that various convergence properties of interest can be described in terms of p-regularity for appropriate choices of p.The dual property, "p-topological ," was introduced and studied in a recent paper by Wilde and Kent [10].
At this point, it is not clear whether, or to what extent, the duality between regular and topological, or p-regular and p-topological, extends to the realm of Cauchy spaces and Cauchy completions.Regular Cauchy completions have been studied extensively (see [8], for example) and p-regular Cauchy completions were investigated in [3].Topological completions (called "diagonal completions" because the Cauchy spaces allowing such completions were characterized by a diagonal condition) have been studied in [5].What is so far missing is a study of p-topological Cauchy completions, which is of course the goal of this paper.
In Section 1, we review some properties of p-topological convergence spaces.Section 2 gives a simple characterization of p-topological Cauchy spaces which allow ptopological completions and shows that, for such spaces, the Wyler completion is the finest p-topological completion.Section 3 gives a diagonal characterization for Cauchy spaces which allow p-topological completions.The Reed family of completions [9] is revisited in Section 4, where it is shown that the Kowalsky completion is the coarsest p-topological completion.For a Cauchy space which allows a p-topological completion, all Reed completions "between" the Kowalsky and Wyler completions also preserve the p-topological property.However, not all p-topological completions are Reed completions and even when p is a topology and (X, Ꮿ) allows a topological completion, a p-topological completion of (X, Ꮿ) need not be topological.Section 5 gives some extension results for p-topological completions, which, for the most part, generalize extension theorems proved for topological completions in [5].

p-topological convergence spaces.
For standard terminology and notation on convergence spaces, the reader is referred to [4] or [6].Let F(X) denote the set of all (proper) filters on a set X; the ultrafilter generated by x ∈ X is denoted by ẋ.If (X, q) is a convergence space, "Ᏺ q-converges to x" is usually written as "Ᏺ q → x"; "cl q " and "I q " denote the closure and the interior operators, respectively.
If X and J are arbitrary sets, a function λ : J → F(X) is called a filter selection function on J.For Ᏺ ∈ F(J), we define kλᏲ to be the filter ∪ F ∈Ᏺ ∩ x∈F λ(x) in F(X); kλᏲ is called the compression of Ᏺ relative to λ.
Let p and q denote two arbitrary convergence structures on a set X. We consider the following axioms: R p,q : Let J be any set, ψ : J → X, and let σ : J → F(X) have the property that σ (y F p,q : Let J be any set, ψ : J → X, and let σ : J → F(X) have the property that σ (y Theorem 1.1 [4].Let (X, q) be a convergence space.
(a) (X, q) is regular if and only if (X, q) satisfies R q,q .
(b) (X, q) is topological if and only if (X, q) satisfies F q,q .
Theorem 1.2 [10].Let (X, q) be a convergence space, p a convergence structure on X.Then (X, q) and p satisfies R p,q if and only if (X, q) is p-regular. (i.e., Ᏺ Definition 1.3 [10].If (X, q) is a convergence space and p a convergence structure on X, then (X, q) is defined to be p-topological if (X, q) and p satisfy F p,q .
A more intuitive characterization of p-topological convergence spaces is obtained via the following definition.Let (X, q) be a convergence space and Ᏺ ∈ F(X).Then ᐂ q (Ᏺ) = {A : I q A ∈ Ᏺ} is called the q-neighborhood filter of Ᏺ.If Ᏺ = ẋ, ᐂ q ( ẋ) = ᐂ q (X) is the q-neighborhood filter at x, obtained by intersecting all the filters which q-converges to x. Theorem 1.4 [10].Let (X, q) be a convergence space and p a convergence structure on X.The following statements are equivalent: (a) (X, q) is p-topological.
A comparison of Theorem 1.4(c) with the definition of p-regularity (Ᏺ q → x implies that cl p Ᏺ q → x) further reveals the duality between "p-regular" and "p-topological".The following corollary to Theorem 1.4 gives an elegant characterization of "topology" in terms of convergence criteria.
Corollary 1.5.A convergence space (X, q) is topological if and only if Proposition 1.6.Let p and q be convergence structures on X.(a) If (X, q) is p-topological, then q ≤ p.(b) If q is a topology, then (X, q) is p-topological if and only if q ≤ p.

Proof. (a) If
Structural properties of p-topological convergence spaces are studied in some detail in [10].A few such results are summarized in the next proposition to convey additional insight into the nature of p-topological spaces.Proposition 1.7.(a) A subspace of a p-topological space is p -topological, where p denotes the restriction of p to the subspace.
(b) Let (X, q) = Π i∈I (X i ,q i ) and (X, p) = Π i∈I (X i ,p i ) be product convergence spaces and assume that each (X i ,q i ) is p i -topological.Then (X, q) is p-topological.
(c) Let C(X) be the lattice of all convergence structures on X.Let = {q i : i ∈ I} ⊆ C(X) and let p ∈ C(X) be such that (X, q i ) is p-topological, for all i ∈ I.If q = inf and s = sup , then (X, q) and (X, s) are both p-topological.

The fine p-topological completion.
A Cauchy space (X, Ꮿ) is a set X with a collection Ꮿ of filters on X satisfying (1) for each x ∈ X, ẋ ∈ Ꮿ; (2 Associated with a Cauchy space (X, Ꮿ) is a convergence structure q Ꮿ on X defined as follows: Ꮿ) is said to be Hausdorff if q Ꮿ -convergent filters have unique limits or, equivalently, if ẋ ∩ ẏ ∈ Ꮿ implies x = y.Two Cauchy filters Ᏺ, Ᏻ ∈ Ꮿ are equivalent if Ᏺ∩Ᏻ ∈ Ꮿ.For a Hausdorff Cauchy space (X, Ꮿ), the set of all Cauchy equivalence classes is denoted by X * ; the natural injection  : X → X * is given by  A completion ((Y , Ᏸ), g) of a Cauchy space (X, Ꮿ) consists of a complete Cauchy space (Y , Ᏸ) along with a dense Cauchy embedding g : (X, Ꮿ) → (Y , Ᏸ).The completion is strict if Ꮽ ∈ Ᏸ implies that there is Ᏺ ∈ Ꮿ such that cl qᏰ (Ᏺ) ≤ Ꮽ, and Hausdorff if (Y , Ᏸ) is Hausdorff.Henceforth, we assume that all Cauchy spaces and Cauchy completions are Hausdorff unless otherwise indicated.
A completion , for all Ᏺ ∈ Ꮿ. Reed [9] proved that every completion is equivalent to one in standard form and ,therefore, unless otherwise stated, we limit our attention to completions in standard form.It is well known that the finest completion of a Cauchy space (X, Ꮿ) is the Wyler completion ((X * , Ꮿ * ), ), generated by The Wyler completion determines a completion functor on the category of Cauchy spaces and Cauchy-continuous maps.Definition 2.1.Let (X, Ꮿ) be a Cauchy space, p a convergence structure on X.
If (X, Ꮿ) is p-topological (respectively, p-regular), it is obvious that the associated convergence structure q Ꮿ is likewise p-topological (respectively, p-regular).It also follows by Proposition 1.6 that if (X, Ꮿ) is p-topological, then q Ꮿ ≤ p.The next proposition describes the upper and lower p-topological modifications of a Cauchy space.Analogous results for p-regular modifications are found in [3, Prop.

and τ p Ꮿ (assuming the latter exists).
To define a p-topological or a p-regular completion of a Cauchy space (X, Ꮿ), it is necessary to first extend p to the completion set X * in an appropriate way.We follow the procedure used to define p-regular completions in [3].Definition 2.3.Let (X, Ꮿ) be a p-topological Cauchy space.Let p * be the finest convergence structure on X * subject to the following conditions: (i) (Ᏺ)p * -converges to (x) in X * if and only if Ᏺ p-converges to x in X; Since q Ꮿ ≤ p, it is clear that p * is a Hausdorff convergence structure on X * .The definition of p * given here differs slightly from that in [3], but the two versions of p * have the same ultrafilter convergence and, consequently, are equivalent for defining either p-regular or p-topological completions.
When referring to a "p-topological Cauchy space" or "p-topological completion", we always assume that p is an arbitrary convergence structure on X (subject, of course, to q Ꮿ ≤ p), unless otherwise specified.Since the p-topological property is hereditary for Cauchy spaces as well as for convergence spaces, it is clear that any Cauchy space which allows a p-topological completion is itself p-topological.
If (X, Ꮿ) is p-topological, it follows directly from Definition 2.3 that the p * -neighborhood filters for points in X * are as follows: ), and if z ∈ X * \(X), then ᐂ p * (z) = ᏹ z ∩ ż, where ᏹ z = ∩z.From these observations, the next lemma follows easily.
In other words, a Cauchy space is cushioned if every equivalence class of nonconvergent Cauchy filters contains a smallest filter (or "cushion").Proposition 2.5.If (X, Ꮿ) is cushioned and p-topological, where p is a topology on X, then p * is a topology on X * .Proof.Let z ∈ X * .By Definition 2.3, it follows that ᐂ p * (z) = (ᐂ p (x)) if z = (x), where x ∈ X, and It is obvious from its construction that the Wyler completion is the largest completion of a given Cauchy space in standard form and hence, the finest Cauchy completion up to equivalence.It is, thus, clear from Theorem 2.7 that it is the finest p-topological completion of any cushioned, p-topological Cauchy space and thus, we refer to it as the fine p-topological completion.Additional p-topological completions are studied in Section 4.

3.
The diagonal axioms D p .topological completion of a Cauchy space (X, Ꮿ) is one for which the convergence structure of the completion space is a topology.In [5], a diagonal condition D is defined for a Cauchy space (X, Ꮿ) which is necessary and sufficient for (X, Ꮿ) to have a topological completion.One might conjecture that a topological completion of (X, Ꮿ) is simply a p-topological completion for which p = q Ꮿ , but this is in fact false, as is shown by an example in the next section.In this section, we define a more general diagonal axiom D p which is necessary and sufficient for a Cauchy space to allow a p-topological completion.
Let (X, Ꮿ) be a Cauchy space and p a convergence structure on X such that q Ꮿ ≤ p.Consider the following axioms.D p : Let J be any set, ψ : J → X * , and let σ : J → Ꮿ be such that σ (y for some Ᏼ ∈ Ꮿ, then kσ Ᏺ ∈ Ꮿ.When p = q Ꮿ , the axiom D p reduces to the axiom D defined in [5]. Theorem 3.1.If (X, Ꮿ) and p satisfy D p , then (X, Ꮿ) is cushioned and p-topological.
Proof.First, we show that (X, Ꮿ) is cushioned.Let z ∈ X * \(X) and let z = {Ᏺ i : i ∈ A} be an indexing of the members of z.Let J = X ∪ A and let ψ : J → X * be defined by Let σ : J → F(X) be defined by Let Ᏺ ∈ z and let Ᏻ be the filter on J with filter base {F ∪ A : Let F ∈ Ᏺ; then K = ψ −1 F is an arbitrary set in .For each pair (Ᏻ,x) ∈ K, choose G (Ᏻ,x) ∈ Ᏻ.Then A = {G (Ᏻ,x) : (Ᏻ,x) ∈ K} is an arbitrary set in kσ .It is easy to see that A ∈ ᐂ p (x), for all x ∈ F .Thus, F ⊆ I p (A), which establishes that Ᏺ ≥ I p (kσ ).Proof.It remains to show that a cushioned, p-topological Cauchy space (X, Ꮿ) satisfies D p .Let J, ψ, and σ be as stated in D p .Let Ᏺ ∈ F(J) be such that there is Proof.Since (X, Ꮿ) satisfies D, q Ꮿ is a topology by [5].By Theorem 2.7, (X * , Ꮿ * ) is (q Ꮿ ) * -topological and it is evident that (q Ꮿ ) * = q Ꮿ * is the convergence structure of the Wyler completion, which is topological by Proposition 2.5.
At this point, we wish to correct an error on [5, p. 263], where it is falsely asserted that the Wyler completion of a Cauchy space satisfying D may fail to be topological, an obvious contradiction to Theorem 3.3.In [5], the fine diagonal completion of a Cauchy space (X, Ꮿ) is defined to be the topological modification of the Wyler completion.This is correct but redundant since, in this case, the topological modification of the Wyler completion is itself.Nonetheless, [5, Thm.2.10] is correct as stated.Theorem 3.4.(a) If a completion ((X * , Ᏸ), ) of a Cauchy space (X, Ꮿ) is topological, then it is also q Ꮿ -topological.
(b) A Cauchy space (X, Ꮿ) has a topological completion if and only if it has a q Ꮿtopological completion and q Ꮿ is a topology on X.
(b) If (X, Ꮿ) has a topological completion the desired implication follows by (a).Conversely, under the given assumptions D q Ꮿ is equivalent to D and so, a topological completion exists by Theorem 3.3.
The converse of Theorem 3.4(a) would assert that, for a Cauchy space satisfying D, every q Ꮿ -topological completion is a topological completion.That this is false is shown in Section 4.

Reed's family of p-topological completions.
Let (X, Ꮿ) be a Cauchy space and Λ the set of all selection functions λ : X * → F(X) which satisfy the following conditions: For every subset Γ ⊆ Λ, Reed [9] defined a completion ((X * , Ꮿ Γ ), ) in standard form and discussed various properties of these completions.We briefly review these definitions.

Theorem 4.4. Let (X, Ꮿ) be p-topological and cushioned and let
To complete the proof, we verify that Corollary 4.6.Let (X, Ꮿ) be p-topological and cushioned.(a) The Kowalsky completion is the coarsest p-topological completion of (X, Ꮿ).(b) The Reed p-topological completions (in standard form) are precisely those of the form ((X * , Ꮿ Γ ), ) where Γ ⊆ Λ.
The Kowalsky completion of a p-topological, cushioned Cauchy space is also called the coarse p-topological completion.The significance of Reed p-topological completions is further demonstrated by the next theorem, which is a partial converse to Theorem 3.4(a).Theorem 4.7.Let (X, Ꮿ) be cushioned and q Ꮿ -topological, where q Ꮿ is a topology.Then any Reed q Ꮿ -topological completion is topological.and (ᏹ z ) λ ∩ ż if z ∈ X * \(X), where ᐂ q c(x) and ᏹ z have filter bases of q Ꮿ -open sets.A simple argument shows that if U ⊆ X is q Ꮿ -open, then U λ is q λ -open.Thus, for every z ∈ X * , ᐂ q λ (z) has a filter base of q λ -open sets and therefore, q λ is a topology and Next, let Γ ⊆ Λ.Then q Γ is the supremum in the lattice of convergence structures on X of {q λ : λ ∈ Λ}.Since "topological" is an initial property for convergence structures, q Γ is a topology and therefore, ((X * , Ꮿ Γ ), ) is a topological completion.Are all the p-topological completions of a cushioned, p-topological Cauchy space Reed completions?The answer is no, as the following example demonstrates.This example also shows that the full converse to Theorem 3.4(a) is false since there are Cauchy spaces (X, Ꮿ) which allow topological completions but for which there exist q Ꮿ -topological completions that are not topological.
Example 4.8.Let X be the set Q of rational numbers and Ꮿ the usual Cauchy structure for Q.We may identify X * with the set R of real numbers, in which case  : Q → R is the identity injection.Let q * = (q Ꮿ ) * denote the topology of the Wyler completion of (X, Ꮿ).Thus, for every y ∈ R, ᐂ q * (y) has a filter base of q * -open sets of the form ((y − , y + ) ∩ Q) ∪{y} for all > 0.
For each n ∈ N, let Ᏻ n be the filter on R generated by the collection of open intervals {(1/n − , 1/n + ) : > 0}.Let s be the pretopology on R with neighborhood filters defined as follows: It is readily apparent that ᐂ s (0) does not have a filter base of s-open sets.So, s is not a topology on R and hence, by Theorem 4.7, does not define a Reed-completion of (X, Ꮿ).However, for every y ∈ R, ᐂ s (y) has a filter base of q * -open sets, from which it follows that ((R, s), ) is a q Ꮿ -topological completion of (X, Ꮿ).
Let (X < Ꮿ) be a Cauchy space which allows a p-topological completion.Since the Kowalsky completion of (X, Ꮿ) is strict, it follows, by Corollary 4.6(a), that every ptopological completion of (X, Ꮿ) is strict.In contrast, a Cauchy space which allows a p-regular completion has a unique strict p-regular completion.There are Cauchy spaces which allow a regular completion but no strict regular completion; for such a Cauchy space, the fine q Ꮿ -regular completion is strict and hence, nonregular.Thus, p-regular and p-topological completions exhibit similar behaviour in this respect.Proposition 4.9.If (X, Ꮿ) allows both a p-regular completion ((X * , Ᏸ), ) and a p-topological completion ((X * , Ᏸ), ) then Ᏸ ≤ Ᏸ .
Proof.Since ((X * , Ꮿ σ ), ) is the coarsest p-topological completion (in standard form), it suffices to show that . The latter filter is in Ᏸ since Ᏸ is p-regular and therefore, Ꮽ ∈ Ᏸ.
The existence of p-regular completions in Reed's completion family has yet to be investigated.

Extension theorems.
Let (X, Ꮿ) and (Y , Ᏸ) be Cauchy spaces, and let f : (X, Ꮿ) → (Y , Ᏸ) be Cauchy-continuous.Let f * : X * → Y * be defined as follows: Then f * is a well-defined function such that the canonical diagram The extension theorem for Wyler completion asserts that in the preceding diagram, f * is Cauchy-continuous when X * and Y * are equipped with the Cauchy structures Ꮿ * and Ᏸ * of the respective Wyler completions.
Theorem 5.1.Let (X, Ꮿ) be a cushioned, p-topological Cauchy space, let (Y , Ᏸ) be a cushioned, p -topological Cauchy space, and let f : (X, Ꮿ) → (Y , Ᏸ) be Cauchycontinuous.Then, for the fine p-topological completion ((X * , Ꮿ * ),  X ) of (X, Ꮿ) and the fine p -topological completion ((Y * , Ᏸ * ),  Y ) of (Y , Ᏸ), all the functions in the following commutative diagram are Cauchy-continuous: p-TOPOLOGICAL CAUCHY COMPLETIONS 507 Theorem 5.1 is an obvious special case of the aforementioned Wyler Extension Theorem.It is worth noting that no assumptions are needed concerning the behavior of f : (X, p) → (Y , p ).This same absence of a necessary relationship between p and p is observed again in Theorem 5.8.
Let f : (X, Ꮿ) → (Y , Ᏸ) be Cauchy-continuous, where (X, Ꮿ) is cushioned and ptopological, and (Y , Ᏸ) is cushioned and p -topological.(No relationship is assumed between the convergence structures p on X and p on Y .)As usual, ((X * , Ꮿ σ ),  X ) denotes the Kowalsky completion of (X, Ꮿ), and we use ((Y * , Ᏸ σ ),  Y ) to denote the Kowalsky completion of (Y , Ꮿ), where σ : Y * → F(Y ) is the appropriate selection function which characterizes this completion in accordance with Proposition 4.2.
Proof.The proof is accomplished by making suitable modifications in the proof of Theorem 5.7.Let q be the convergence structure on X * induced by C and q the convergence structure on Y * induced by Ꮿ.Let Ꮽ q → z in X * and n ∈ N. Since q σ ≤ q, Ꮽ qσ → z and so, there is Ᏻ ∈ Ꮿ such that Ꮽ ≥ Ᏻ σ ∩ ż; furthermore, as in the preceding proof,  Y f (Ᏻ) q σ → f * (z).From this, it follows that  Y f (Ᏻ) q → f * (z), since a filter on Y * containing  Y Y exhibits the same convergence relative to any completion of (Y , Ᏸ) in standard form.Thus, f * (cl n q Ꮽ) ≥ f * (cl n qσ Ꮽ) ≥ f * (cl n+1 qσ  X (Ᏻ)) ≥ cl n+1 q σ  Y (f (Ᏻ)) = cl n+1 q  Y f (Ᏻ), where the last equality follows by Corollary 5.5.This completes the proof of the theorem.

Theorem 3 . 2 .
(X, Ꮿ) allows a p-topological completion if and only if (X, Ꮿ) and p satisfy D p .
from which it follows that ᏹ z has a filter base of p-open sets (where A ⊆ X is p-open if and only if I p A = A).It follows easily that ᐂ p * (Ꮽ) ≥ ᐂ p * (z) = ᏹ z ∩ ż, where the latter filter has a base of p * -open sets.Thus, ᐂ p * (Ꮽ) ∈ Ꮿ * and thus, (X * , Ꮿ * ) is p * -topological.