p-Topologicalness — A Relative Topologicalness in >-Convergence Spaces

In this paper, p-topologicalness (a relative topologicalness) in >-convergence spaces are studied through two equivalent approaches. One approach generalizes the Fischer’s diagonal condition, the other approach extends the Gähler’s neighborhood condition. Then the relationships between p-topologicalness in >-convergence spaces and p-topologicalness in stratified L-generalized convergence spaces are established. Furthermore, the lower and upper p-topological modifications in >-convergence spaces are also defined and discussed. In particular, it is proved that the lower (resp., upper) p-topological modification behaves reasonably well relative to final (resp., initial) structures.


Introduction
The theory of convergence spaces [1] is natural extension of the theory of topological spaces.The topologicalness is important in the theory of convergence spaces since it mainly researches the condition of a convergence space to be a topological space.Generally, two equivalent approaches are used to characterize the topologicalness in convergence spaces.One approach is stated by the well-known Fischer's diagonal condition [2], the other approach is stated by Gähler's neighborhood condition [3].In [4], by considering a pair of convergence spaces (X, p) and (X, q), Wilde and Kent investigated a kind of relative topologicalness, called p-topologicalness.When p = q, p-topologicalness is equivalent to topologicalness in convergence spaces.They also defined and discussed the lower and upper p-topological modifications in convergence spaces.Precisely, for a pair of convergence spaces (X, p) and (X, q), the lower (resp., upper) p-topological modification of (X, q) is defined as the finest (resp., coarsest) p-topological convergence space which is coarser (resp., finer) than (X, q).Similarly, a topological modification of (X, q) is defined as the finest topological convergence space which is coarser than (X, q).Lattice-valued convergence spaces are common extension of convergence spaces and lattice-valued topological spaces.It should be pined out that lattice-valued convergence spaces are established on the basis of fuzzy sets.However, the lattice structure is used to replace the unit interval [0, 1] as the truth table for membership degrees.In recent years, two kinds of lattice-valued convergence spaces received much attention: (1) the theory of stratified L-generalized convergence spaces based on L-filters, which is initiated by Jäger [5] and then developed by many researchers [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]; and (2) the theory of -convergence spaces based on -filters, which is investigated by Fang [26] in 2017.The topologicalness in stratified L-generalized convergence spaces was studied by  and Li [30,31], the p-topologicalness and p-topological modifications in stratified L-generalized convergence spaces were discussed by Li [32,33].
The topologicalness in -convergence spaces was researched by Fang [26] and Li [34].In this paper, we shall consider the p-topologicalness and p-topological modifications in -convergence spaces.
The contents are arranged as follows.Section 2 recalls some basic notions as preliminary.Section 3 discusses the p-topologicalness in -convergence spaces by generalized Fischer's diagonal condition and generalized Gähler's neighborhood condition, respectively.Then the relationships between p-topologicalness in -convergence spaces and p-topologicalness in stratified L-generalized convergence spaces are established.Section 4 focuses on p-topological modifications in -convergence spaces.The lower and upper p-topological modifications in -convergence spaces are defined and discussed.Particularly, it is proved that the lower (resp., upper) p-topological modification behaves reasonably well relative to final (resp., initial) structures.

Preliminaries
Let L be a complete lattice with the top element and the bottom element ⊥.For a commutative quantale, we mean a pair (L, * ) such that * is a commutative semigroup operation on L with the condition In the following, we list the usual properties of * and → [35].
We call (L, * ) to be a meet continuous lattice if the complete lattice L is meet continuous [36], that is, (L, ≤) satisfies the distributive law: a ∧ ( i∈I b i ) = i∈I (a ∧ b i ), for any a ∈ L and any directed subsets L is said to be continuous if (L, ≤) is a continuous lattice [36], that is, for any nonempty family {a j,k |j ∈ J, k ∈ K(j)} in L with {a j,k |k ∈ K(j)} is directed for all j ∈ J, the identity holds, where N is the set of all choice functions on J with values h(j) ∈ K(j) for all j ∈ J. Obviously, continuity implies meet-continuity for L.
In this article, unless otherwise stated, we always assume that L = (L, * ) is a commutative, integral, and meet continuous quantale.
A function µ : X → L is called an L-fuzzy set in X, and all L-fuzzy sets in X is denoted as L X .The operations ∨, ∧, * , → on L can be translated pointwisely onto L X .Said precisely, for any µ, ν ∈ L X and any {µ We don't distinguish between a constant function and its value because no confusion will occur.Let f : X −→ Y be a function.We define f → : Let µ, ν be L-fuzzy sets in X.The subsethood degree [37][38][39][40] of µ, ν, denoted by S X (µ, ν), is defined by

-Filters and Stratified L-Filters
A filter on a set X is an upper set of (2 X , ⊆) (2 X denotes the power set of X) wich is closed for finite meets and does not contain the empty set.The conception of filter has been generalized to the fuzzy setting in two methods; prefilters (or -filters more general) and L-filters.Both prefilters ( -filters) and L-filters play important roles in the theory of fuzzy topology, see [26,27,34,35,[41][42][43][44].

Definition 1 ([35]
).A nonempty subset F ⊆ L X is called a -filter on the set X whenever: The set of all -filters on X is denoted as F L (X).

Definition 2 ([35]
).A nonempty subset B ⊆ L X is called a -filter base on the set X provided: Each -filter base generates a -filter F B defined by Example 1 ([26,45]).Let f : X −→ Y be a function, F ∈ F L (X) and G ∈ F L (Y).Then (1) The family { f → (λ)|λ ∈ F} forms a -filter base on Y, and the -filter f ⇒ (F) generated by it is called the image of F under f .It is easily seen The set of all stratified L-filters on X is denoted as Conversely, for each tightly stratified L-filter F on a set X, the family Proof.For any λ ∈ L X and any j ∈ J, note that {S X (µ j , λ)

-Convergence Spaces and Stratified L-Generalized Convergence Spaces
Definition 3. A -convergence structure [26] on a set X is a function q : F L (X) −→ 2 X satisfying The category whose objects are -convergence spaces and whose morphisms are continuous functions will be denoted by -CS.This category is topological over SET [26,46].For a given source (X f i −→ (X i , q i )) i∈I , the initial structure [47], q on X is defined by For a given sink ((X i , q i ) f i −→ X) i∈I , the final structure, q on X is defined as Thus, when X = ∪ i∈I f i (X i ), the final structure q can be simplified as For a nonempty set X, we use (X) to denote all -convergence structures on X.For p, q ∈ (X), we say that q is finer than p, or p is coarser than q, denoted by p ≤ q for short, if the identity id X : ( [26,47] that ( (X), ≤) forms a completed lattice, and the discrete (resp., indiscrete) structure δ (resp., ι) is the top (resp., bottom) element of (T(X), ≤), where δ is given by F Definition 4. (Jäger [5] and Yao [25]) A stratified L-generalized convergence structure on a set X is a function Let (X, q) be a -convergence space.We define lim q : F s L (X) −→ L X as ).It follows that (X, lim q ) is a stratified L-generalized convergence space.
Remark 1.When L = {⊥, }, both -convergence spaces and stratified L-generalized convergence spaces all reduce to convergence spaces.Therefore, these two kinds of lattice-valued convergence spaces are all natural extensions of convergence spaces.

p-Topologicalness in -Convergence Spaces
In this section, we shall discuss the p-topologicalness in -convergence spaces by generalized Fischer's diagonal condition and generalized Gähler's neighborhood condition, respectively.We also try to establish the relationships between p-topologicalness in -convergence spaces and p-topologicalness in stratified L-generalized convergence spaces.

p-Pretopologicalness in -Convergence Spaces
Let (X, p) be a -convergence space.Then for any x ∈ X, the -filter is called the -neighborhood with respect to p at x. Then the family U p := {U p (x)} x∈X is called the -neighborhood system generated by (X, p) [26].It is easily seen that if p, p ∈ T(X) and p ≤ p then U p (x) ⊆ U p (x) for any x ∈ X.
In the following, we shorten a pair of -convergence spaces (X, p) and (X, q) as (X, p, q).It is easy to check that the following conditions are equivalent: Definition 5. Assume that (X, p, q) is a pair of -convergence spaces.Then q is said to be p-pretopological if it fulfills either of the above three conditions.Remark 2. When p = q, p-pretopologicalness is precise the pretopologicalness in [26].In this case, it is observed easily that the "=⇒" in p-(TP2) can be replaced with "⇐⇒".In the following, when p = q, we omit the prefix "p" in symbols p-(TP1)-p-(TP3).This simplification is also used for the subsequent p-topological conditions.Proposition 1.A -convergence structure q on X is pretopological iff it is p-pretopological for any q ∈ (X) with q ≤ p.
Proof.Let (X, q) be pretopological and q ≤ p. Then by q ≤ p we have U p (x) ⊇ U q (x) for any x ∈ X.By pretopologicalness of q we get that U q (x) The converse implication is obvious.
The following example shows there is no p-pretopologicalness implies pretopologicalness in general.
For x, y ∈ X, it is easily seen that the subsets F x , F y of L X defined by Obviously, q satisfies p-(TP3).But q is not pretopological since we have no U q (z) q −→ z.

p-Topologicalness in -Convergence Spaces
At first, we fix the notions of diagonal -filter and neighborhood -filter to state p-topologicalness.Let J, X be any sets and φ : J −→ F L (X) be any function.Then a function φ : L X → L J is defined as For all F ∈ F L (J), it is proved that a subset of L X defined by kφF := {λ ∈ L X | φ(λ) ∈ F} is a -filter, called diagonal -filter of F under φ [26].In addition, for any λ, µ ∈ L X , it was proved in [26] that S X (λ, µ) ≤ S J ( φ(λ), φ(µ)).

Definition 6 ([34]
).Let (X, p) be a -convergence space and U p : X −→ F L (X) be the -neighborhood system generated by (X, p).Then for each F ∈ F L (X), the -filter U p (F) := kU p F, is called neighborhood -filter of F w.r.t.p.
Let N be the set of natural numbers including 0. Let (X, p) be a -convergence space and F ∈ F L (X).For any n ∈ N, we define U 0 p (F) = F, and if U n p (F) has been defined, then we define the n + 1 th iteration of the neighborhood -filter of F inductive by U n+1 p (F) = U p (U n p (F)).
Proposition 2. Let (X, p) be a -convergence space, n ∈ N and F, G ∈ F L (X).
Proof.It is obvious.
Definition 7. Let f : (X, q) −→ (Y, p) be a function between -convergence spaces.Then f is said to be an interior function if f → ( U q (λ)) ≤ U p ( f → (λ)) for all λ ∈ L X .
Proposition 3. Let f : (X, q) −→ (Y, p) be a function between -convergence spaces and F ∈ F L (X). ( Fixing λ ∈ L Y , we get (2) We check only the inequalities for n = 1.Let f be an interior function.For each λ ∈ U q (F), i.e., U q (λ) Now, we tend our attention to p-topologicalness.We say a pair of -convergence spaces (X, p, q) satisfy the Gähler -neighborhood condition if Definition 8. Let (X, p, q) be a pair of -convergence spaces.Then q is called p-topological if the condition p-(TG) is satisfied.
Remark 3. When L = {⊥, }, the condition p-(TG) is precise the Gähler neighborhood condition in [4], which is used to define p-topological convergence spaces.Therefore, our p-topologicalness is a natural extension of crisp p-topologicalness.
We say a pair of -convergence spaces (X, p, q) satisfy the Fischer -diagonal condition if p-(TF): Let J, X be any sets, ψ : J −→ X, and φ : J −→ F L (X) such that φ(j) p −→ ψ(j), for each j ∈ J.
Then for each F ∈ F L (J) and each x ∈ X, ψ ⇒ (F) Restricting J = X and ψ =id in p-(TF), we obtain a weaker condition p-(TK).When p = q, p-(TF) is precise the Fischer -diagonal condition (TF), and p-(TK) is precise the Kowalsky -diagonal condition (TK) in [26].
Remark 4. Let L, X and F z (z ∈ X) be defined as in Example 3. Let q be defined as F q −→ z for any F ∈ F L (X) and any z ∈ X, and let p be defined as F p −→ z ⇐⇒ F ⊃ F ⊥ .Then (X, p, q) is a pair of -convergence spaces.
Obviously, the axiom p-(TF) is satisfied.But p does not fulfill the axiom (TP1) since U p (z) = F ⊥ p −→ z.Thus this example shows that p-(TF) does not imply (TP1) of (X, p) generally.Therefore, we guess that the additional condition (TP1) in the above corollary can not be removed.
The following theorem shows that if we restricting the lattice-context slightly, p-topologicalness can be described by Fischer -diagonal condition p-(TF).Theorem 1.Let (X, p, q) be a pair of -convergence spaces.Then p-(TG)=⇒p-(TF), and the converse inclusion holds if L is continuous.
The following theorem shows that for pretopological -convergence spaces, p-topologicalness can be described by Fischer -diagonal condition p-(TF).Theorem 2. Let (X, p, q) be a pair of -convergence spaces and (X, p) be pretopological.Then p-(TF)⇐⇒p-(TG).
Proof.Most of the proof can copy that of Theorem 1.We only check that µ∈F for any λ ∈ L X in p-(TF)=⇒p-(TG).Indeed, since p is pretopological then U p (y) ∈ I y for any y ∈ X.

Remark 5.
The above corollary is one of the main results in [34].Based on this equivalence, it was proved that -convergence spaces with (TF) or (TG) characterize precisely the conical L-topological spaces in [44].
The following theorem shows that p-topologicalness is preserved under initial constructions.
Theorem 3. Let {(X i , q i , p i )} i∈I be pairs of -convergence spaces with each q i being p i -topological.If q (resp., p) is the initial structure on X relative to the source (X Then by definition of q, we have f ⇒ i (F) The next theorem shows that p-topologicalness is preserved under final constructions with some additional conditions.Theorem 4. Let {(X i , q i , p i )} i∈I be pairs of -convergence spaces with each q i being p i -topological.Let q (resp., p) be the final structure on X w.r.t.The sink ((X i , q i ) and each f i : (X i , p i ) −→ (X, p) is an interior function, then (X, q) is p-topological.
Proof.Let F q −→ x.Then by definition of q, there exists i ∈ Then it follows that U p (F) q −→ x.By Theorem 1 we get that q is p-topological.
From Theorem 3 and Theorem 4, we conclude easily the following corollary.It will tell us that p-topologicalness is preserved under supremum and infimum in the lattice (X).Corollary 3. Let {q i |i ∈ I} ⊆ (X) and p ∈ (X) such that each (X, q i ) is p-topological.Then both (X, inf{q i } i∈I ) and (X, sup{q i } i∈I ) are all p-topological.

On the Relationship between p-Topologicalness in -Convergence Spaces and in Stratified L-Generalized Convergence Spaces
Let J, X be any set and Φ : J −→ F s L (X) be any function.Then a function Φ : L X → L J is defined as ∀λ ∈ L X , ∀j ∈ J, Φ(λ)(j) = Φ(j)(λ).For all F ∈ F s L (J), it is proved that the function KΦF : L X −→ L defined by ∀λ ∈ L X , KΦF (λ) = F ( Φ(λ)) is a stratified L-filter, which is called the diagonal L-filter of F under Φ [27,30].
Let (X, lim p ) be a stratified L-generalized convergence space.For any α ∈ L, x ∈ X, let U α p (x) = {F : lim p F (x) ≥ α}.Take Φ = U α p : X −→ F s L (X), then for each F ∈ F s L (X), the stratified L-filter U α p (F ) := kU α p F is called α-level neighborhood L-filter of F w.r.t.lim p [29].We say a pair of stratified L-generalized convergence spaces (X, lim p , lim q ) satisfy the Fischer L-diagonal condition if p-(LF): Let J, X be any sets, Ψ : J −→ X and Φ : J −→ F s L (X) be functions.
We say a pair of stratified L-generalized convergence spaces (X, lim p , lim q ) satisfy the Gähler L-neighborhood condition if p-(LG): ∀α ∈ L, ∀F ∈ F s L (X), α * lim q F ≤ lim q U α p (F ).It was proved in [32] that p-(LF)⇐⇒ p-(LG).Definition 9 ([32]).Let (X, lim p , lim q ) be a pair of stratified L-generalized convergence spaces.Then lim q is called p-topological if the condition p-(LF) or p-(LG) is satisfied.Lemma 2. Let φ : J −→ F L (X) be any function and F ∈ F L (J).Then