p-topological and p-regular: dual notions in convergence theory

The natural duality between"topological"and"regular,"both considered as convergence space properties, extends naturally to p-regular convergence spaces, resulting in the new concept of a p-topological convergence space. Taking advantage of this duality, the behavior of p-topological and p-regular convergence spaces is explored, with particular emphasis on the former, since they have not been previously studied. Their study leads to the new notion of a neighborhood operator for filters, which in turn leads to an especially simple characterization of a topology in terms of convergence criteria. Applications include the topological and regularity series of a convergence space.


Introduction.
In 1990, G. Richardson and one of the authors introduced the notion of p-regular convergence space, [6], defined as follows: If q and p are convergence structures on a set X, then the space (X, q) is p-regular if cl p Ᏺ q → x whenever Ᏺ q → x, where "cl p " is the p-closure operator. Clearly p-regularity is equivalent to regularity when p = q. By varying p, one can characterize various convergence properties in terms of p-regularity (see [6,7]).
More recently, Kent and Richardson [7] developed some ideas and results due to Kowalsky [8], Cook and Fischer [1], and Biesterfeldt [2] to give convergence characterizations of the properties "topological" and "regular" so as to reveal a fundamental duality between these notions. These characterizations made use of "diagonal" axioms F and R which are in a natural way dual to each other. (It should be noted that the axiom called R in this paper was called DF in [7].) In this paper, we begin by proving the p-regularity of a convergence space (X, q) also has a "diagonal" characterization in terms of an axiom we call R p,q , which is obtained by making a minor alteration in the axiom R. We then use the dual axiom F p,q to define (and introduce) the dual notion of a "p-topological convergence space." Our goal is two-fold. We wish to study and develop this new concept of a ptopological convergence space, while simultaneously exploring the duality alluded to in the title of the paper. The approach based on duality is most useful in examining the structural behavior of p-topological and p-regular spaces as well as their upper and lower modifications. This approach is adopted in Sections 1 and 4. In Section 2, we study some aspects of p-topological spaces which do not have obvious analogues in the setting of p-regular spaces. Section 3 introduces the "neighborhood operator for filters" which seems to be "tailor-made" for the study of p-topological spaces and is used extensively in Section 4. The characterization of p-topological spaces, given in Theorem 3.2, yields a corollary which gives a simple and elegant characterization of a topology in terms of convergence criteria.
As is shown in Section 2 of this paper and also in [6,7], both of the properties "p-topological" and "p-regular" can be adapted to characterize various convergence and topological concepts and, thereby, reveal underlying relationships between them. Other applications of these notions include the regularity and topological series of a convergence space which are discussed briefly in Section 5.
1. The Axioms F p,q and R p,q . For standard notation and terminology pertaining to convergence spaces, the reader is referred to [7]. In particular, F(X) denotes the set of all filters on a set X, U(X) the set of all ultrafilters on X, and C(X) the complete lattice of all convergence structures on X (with the discrete topology as the greatest element). Letẋ denote the fixed ultrafilter on X generated by x ∈ X.
If (X, q) is a convergence space and J an arbitrary set, let Ᏺ ∈ F(J) and let σ : J → F(X) be an arbitrary "selection function." We define κσ Ᏺ to be the filter ∪ F ∈Ᏺ ∩ x∈F σ (x) in F(X); κσ Ᏺ is called the compression of Ᏺ relative to σ .
We, next, define two axioms pertaining to two convergence structures p, q on a set X. F p,q : Let J be any set, ψ: J → X, and let σ : J → F(X) have the property that σ (y) R 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. Let (X, q) be a convergence space and p ∈ C(X). Then (X, q) is pregular if and only if p and q satisfy R p,q .
Assume that Ᏺ q → x. We define a filter Ᏼ ∈ F(J) as follows: for each F ∈ Ᏺ, let H F = {(Ᏻ,y) ∈ J: F ∈ Ᏻ}, and let Ᏼ be the filter on J generated by for every y ∈ F . Then σ (y) p → ψ(y), for every y ∈ F , which implies that ψ(y) ∈ cl p ( y∈F A y ) holds for every y ∈ F , and so ψ(F ) ⊆ cl p ( y∈F A y ). Since y∈F A y is a basic set in κσ Ᏺ, the claim is verified. By p-regularity, cl p (κσ Ᏺ) If (X, q) is a convergence space and p ∈ C(X), then (X, q) is defined to be p-topological if (X, q) and p satisfy the axiom F p,q . Note that, by Theorem 1.1, (X, q) is p-regular if and only if (X, q) and p satisfy R p,q . Since F p,q and R p,q are dual to each other, "p-topological" and "p-regular" are likewise dual properties. In the special case where p = q, F q,q and R q,q are denoted by F and R, respectively.
is regular if and only if (X, q) satisfies R.
Proof. The first assertion is proved in [6], the second by combining results from [2,1].
It follows from Theorem 1.2 that "p-topological" generalizes "topological" in the same way that "p-regular" generalizes "regular". In the next theorem, F p,q and R p,q are applied directly to determine the behavior of these properties relative to initial constructs.

Theorem 1.3. Initial structures.
(i) Let {(X i ,q i ): i ∈ I} be a set of spaces together with a set of convergence structures p i which satisfies F p i ,q i , for all i ∈ I. Let X be a set and let f i : X → X i be a mapping, for each i ∈ I. If q is the initial structure on X relative to the families {(X i ,q i ): i ∈ I} and {f i : i ∈ I}, and p is the initial structure on X relative to {p i : i ∈ I} and {f i : i ∈ I}, then (X, q) and p satisfy F p,q .
, for all i ∈ I. Let J be a set and ψ: J → X and σ : J → F(X) have the property that σ (j) p → ψ(j) for all j ∈ J. Define σ i (j) and ψ i (j) so that σ i (j) = f i (σ (j)) and ψ i (j) = f i (ψ(j)) for all j ∈ J and i ∈ I, respectively. Thus, for all i ∈ I, by the property of q being the initial structure of all the q i . Thus, f i (κσ Ᏺ) = κσ i Ᏺ q i → f i (x) for all i ∈ I by the property F p i ,q i . Hence, κσ Ᏺ q → x by the definition of q and this implies that (X, q) and p satisfy F p,q .
(ii) This proof is essentially the same as that of (i).
Corollary 1.6. Let X be a set and let Λ = {q i : i ∈ I} and Γ = {p i : i ∈ I} be subsets of C(X). Let q = sup Λ and p = sup Γ . If (X i ,q i ) is p i -topological (respectively, p i -regular) for each i ∈ I, then (X, q) is p-topological (respectively, p-regular).
Before proving the analogue of Theorem 1.3 for final structures, we give a simpler characterization for p-topological spaces which makes use of the p-interior operator Ᏽ p .
Thus, to show that (X, q) satisfies the given condition, it suffices to show that Ᏺ ≥ Ᏽ p (κσ Ᏼ).
and we obtain the desired conclusion that Ᏺ ≥ Ᏽ p (κσ Ᏼ).
Conversely, let J, ψ, σ , and Ᏺ be as in F p,q and let ψ(Ᏺ) Let f : (X, q) → (Y , p) be a function between convergence spaces. We define f to be an interior map if f (Ᏽ p (A)) ⊆ Ᏽ p (f (A)) holds for all A ⊆ X, and a closure map if cl p (f (A)) ⊆ f (cl q (A)) holds for all A ⊆ X. Closure maps were introduced in [6], where they were found to be useful in the study of p-regularity.
The proof of (ii) is similar.
The final result of this section, which follows immediately from Corollaries 1.6 and 1.11, asserts that for a fixed convergence structure p on X, both of the properties "p-topological" and "p-regular" are preserved under arbitrary infima and suprema in the lattice C(X). , p-regular), for all i ∈ I. Let q = inf Λ and r = sup Λ. Then both (X, q) and (X, r ) are p-topological (respectively, p-regular).
2. More on p-topological spaces. In Section 1, we observed that p-topological and p-regular properties exhibit essentially the same structural behavior. Now, we gain some additional insight into the behavior of p-topological spaces by making use of Theorem 1.7. The first result of this section gives a simple characterization of pretopological spaces which are p-topological. Theorem 2.1. Let (X, q) be a pretopological space and p ∈ C(X).
( Let X = R be the set of real numbers, and let τ denote the usual topology on R. Note that τ ∪{0} is a base for a topology p on R, where τ < p and ᐂ p (x) = ᐂ τ (x), for all x ≠ 0, whereas ᐂ p (0) =0. Let q be the pretopology on R defined by ᐂ q (x) = ᐂ τ (x) for x ≠ 0 and ᐂ q (0) = ᐂ τ (0) ∨Q (whereQ is the filter of oversets of the set Q of rational numbers). Note that τ < q < p. Then ᐂ q (x) q → x, but Ᏽ p ᐂ q (0) =0 ≠ ᐂ q (0), so by Theorem 2.1(i), (X, q) is not p-topological. Proof. If (X, q) is p-topological, then p ≤ q follows from Theorem 2.1(ii). Conversely, if q ≤ p, then Ᏽ p ᐂ q (x) = ᐂ q (x) follows because q is a topology, and so, the conclusion follows from Theorem 2.1(i).
The preceding example shows that Corollary 2.3 does not hold under the weaker condition that q is pretopological.
Note that if (X, q) is p-topological, then (X, q) is obviously p -topological for any p ≥ p. Clearly, every convergence space is δ-topological, where δ denotes the discrete topology.
Given a convergence space (X, q), let ρq denote the finest completely regular topology on X coarser than q, and let ωq be the finest completely regular topology on X coarser than q.
Let (X, q) be a topological space, and let q be the topology on X generated by Ꮾ q = {X}∪{U ⊆ X: U ∈ q and U ⊆ K for some q-compact subset K of X}.

Theorem 2.7. A T 1 topological space (X, q) is locally compact if and only if (X, q)
is q -topological.
Proof. Let (X, q) be locally compact and x ∈ X. Let U ∈ ᐂ q (x) be q-open. By local compactness, there is a compact set A ∈ ᐂ q (x). Let V be a q-open set such that V ⊆ U ∩A. Then V is q -open. So, Ᏽ q ᐂ q (x) = ᐂ q (x), which implies, by Theorem 2.1(i), that (X, q) is q -topological.
A related theorem characterizing local compactness in terms of p-regularity is the following result, which is a direct corollary of [6, Thm. 3.1].
Theorem 2.8. Let (X, q) be a T 1 convergence space. Let p be the topology on X having as a base of closed sets all the nonempty subsets of q-compact sets. Then (X, q) is locally compact if and only if (X, q) is p-regular.

The neighborhood operator for a filter.
In this section, we introduce a new filter notion which is essentially dual to the "closure of a filter," thereby obtaining another characterization of "p-topological" which further illustrates its duality with "p-regular".
Proof. It is clear that ᐂ q Ᏺ is a filter on X such that Ᏽ q ᐂ q Ᏺ ≤ Ᏺ. If Ᏻ is any filter on X such that Ᏽ q Ᏻ ≤ Ᏺ, then G ∈ Ᏻ implies Ᏽ q G ∈ Ᏺ, and, hence, G ∈ ᐂ q Ᏺ.
If Ᏺ =ẋ, it is obvious from the definition that ᐂ qẋ = ᐂ q (x) is the q-neighborhood filter at x.
The corresponding dual characterization for a p-topological space is the following.

Theorem 3.2. A convergence space (X, q) is p-topological if and only if
Proof. Let(X, q)be p-topological and Ᏺ q → x. By Theorem 1.7, there isᏳ Conversely if the condition holds, we can set Ᏻ = ᐂ p Ᏺ in Theorem 1.7, and, thus, (X, q) is p-topological.

Corollary 3.3. A convergence space (X, q) is topological if and only if
The q-neighborhood filter of a filter can also be described by means of the compression operator for filters defined in Section 1. Proof. Let A ∈ ᐂ p (Ᏺ). Then Ᏽ p (A) ∈ Ᏺ. If F = Ᏽ p (A), then for each x ∈ F, A ∈ ᐂ p (x), and so A = x∈F V x , where each V x = A, is a basic set in κσ Ᏺ. Conversely, let A ∈ κσ Ᏺ. Then A contains a basic set of the form B = y∈F V y , where F ∈ Ᏺ and V y ∈ ᐂ p (y), for all y ∈ F . To show that A ∈ ᐂ p (Ᏺ), it suffices to show that Let (X, q) be a convergence space and Ᏺ ∈ F(X). For any n ∈ N, the set of natural numbers, the nth iterations of the closure and neighborhood operators for a filter Ᏺ are given inductively by: The next two propositions summarize (without proof) some additional elementary properties of the neighborhood operator for filters.
(i) If f is continuous, then f ᐂ n q Ᏺ ≥ ᐂ n p f (Ᏺ). (ii) If f is an interior map, then f ᐂ n q Ᏺ ≤ ᐂ n p f (Ᏺ).

Lower and upper modifications.
It was established in Corollary 1.12 that each of the properties p-topological and p-regular is preserved under both infima and suprema in the lattice C(X). Since an indiscrete space is both p-topological and pregular for any choice of p, we immediately obtain the following. Proposition 4.1. Let (X, q) be a convergence space and p ∈ C(X).
(i) There is a finest p-topological convergence structure τ p q on X coarser than q. (ii) There is a finest p-regular convergence structure r p q on X coarser than q.
The structures τ p q and r p q are called the lower p-topological and lower p-regular modifications of q, respectively. The dual relationship between these concepts is evident in the next theorem.  q → x such that Ᏺ ≥ ᐂ n p (Ᏻ), for some n ∈ N. One may easily verify that q is a convergence structure. If Ᏺ q → x, then Ᏺ ≥ ᐂ n p (Ᏺ) for any n ∈ N, and so Ᏺ q → x. Thus, q ≤ q. To show that q is p-topological, let Ᏺ (ii) If both of τ p q and τ p q exist and f : (X, p) → (X ,p ) is an interior map, then f : (X, τ p q) → (X, τ p q ) is continuous.
(iii) If both of r p q and r p q exist and f : (X, p) → (X ,p ) is a closure map, then f : (X, r p q) → (X, r p q ) is continuous.
Proof. Those results pertaining to p-regular structures have been proved in [6]. Those pertaining to p-topological structures can be proved analogously using Theorems 4.2(i) and 4.4(i), along with Proposition 3.6.
The next two theorems show that the lower modifications behave reasonably well relative to final structures, whereas the upper modifications exhibit comparable behavior relative to initial structures.
where the last inequality follows by Proposition 3.6. Thus, Ᏺ τp q → x.
The proof of (ii) is the similar.
To avoid needless repetition, we state the next three corollaries to Theorem 4.6 only for the lower p-topological modifications. Analogous results obviously hold for the lower p-regular modifications as well.  Let (X, q) = i∈I (X i ,q i ) be a disjoint sum of convergence spaces, and let p ∈ C(X) be such that each g i : (X i ,p i ) → (X, p) is an interior-preserving map, where g i : X i → X is the canonical injection. Then (X, τ p q) = i∈I (X i ,τ p i q i ).  (i) If τ p i q i exists for all i ∈ I and each f i : (X, p) → (X i ,p i ) is an interior map, then τ p q exists and is the initial structure on X induced by {f i : i ∈ I} and {(X i ,τ p i q i ): i ∈ I}.
(ii) If r p i q i exists for all i ∈ I and each f i : (X, p) → (X i ,p i ) is a closure map, then r p q exists and is the initial structure on X induced by {f i : i ∈ I} and {(X i ,r p i q i ): i ∈ I}.
Proof. (i) To show τ p q exists, it suffices, by Theorem 4.3, to show that if Ᏺ ≥ ᐂ n p (x) for some x ∈ X and n ∈ N, then Ᏺ , and since each τ p i q i exists by assumption, The remainder of the proof of (i) is straight-forward and is omitted. The proof of (ii) exactly parallels that of (i).
The corollaries of Theorem 4.10, like those of Theorem 4.6, are stated only for the upper p-topological modifications. The corresponding results involving upper pregular modifications can be supplied by the reader.
Corollary 4.11. Let (X, q) be a subspace of (X ,q ) and let p ∈ C(X ). Also let (X, p) be a subspace of (X ,p ) and assume that τ p q exists. If X is p -open, then (X, τ p q) is a subspace of (X ,τ p q ).
Proof. Since X is p -open in (X ,p ), the identity map from (X, p) into (X ,p ) is a continuous interior map, and so, the conclusion follows from Theorem 4.10(ii). Corollary 4.12. Let (X, q) = Π i∈I (X i ,q i ), let p i ∈ C(X i ) be such that τ p i q i exists for each i ∈ I, and let p ∈ C(X) be such that the ith projection map π i : (X, p) → (X i ,p i ) is continuous interior map for all i ∈ I. Then τ p q exists, and (X, τ p q) = Π i∈I (X i , τ p i q i ). Corollary 4.13. Let X be a set, Λ = {q i : i ∈ I} ⊆ C(X), and let p ∈ C(X) be such that τ p q i exists for all i ∈ I. If q = sup Λ, then τ p q exists and τ p q = sup{τ p q i : i ∈ I}.

The topological series of a convergence space.
If (X, q) is a convergence space, it is well known that there is a finest topology τq coarser than q and a finest regular convergence structure r q coarser than q. These are the topological and regular modifications of q. However, neither τq-convergence nor r q-convergence can be described directly in terms of q-convergence. Consequently, descending ordinal series have been devised to "bridge the gap" between q and these two lower modifications.
The regularity series (r α q), introduced in [4] and studied also in [5], can be easily characterized by means of the lower p-regular modification for an arbitrary ordinal number α as follows: r α q = r pα q, where p 0 = δ, p 1 = q, p α = r α−1 q if α − 1 exists, and p α = inf {r β q: β < α} if α is a limit ordinal. The least ordinal α for which r α q = r α+1 q is called the length of the regularity series and is denoted by R q. It is easy to verify that r α q = r q if and only if α ≥ R q.
The decomposition series (π α q), introduced in [3], is a descending ordinal sequence of pretopologies terminating in τq. Just as the regularity series gives an ordinal measure of how "non-regular" a given convergence space is, so likewise does the decomposition series measure how "non-topological" the given space is. However, the construction of the regularity and decomposition series are fundamentally so different that interactions or comparisons between them are difficult to find or interpret. The existence of the lower p-topological modification and its dual relationship to the lower p-regular modification provide means for constructing a new descending ordinal sequence called the topological series (τ α q) of (X, q) which, like the decomposition series, stretches between q and τq. Following the preceding description of the regularity series, we define: τ α q = τ pα q, where p 0 = δ, p 1 = q, p α = τ α−1 q if α − 1 exists, and p α = inf {τ βq : β < α} if α is a limit ordinal.The resulting topological series is the exact dual of the regularity series. It can be shown that the length of the topological series cannot exceed that of the decomposition series. Additional results pertaining to these and other related ordinal series will be published later.