Regularity of fuzzy convergence spaces

Lingqiang Li; Qiu Jin; Bingxue Yao

doi:10.1515/math-2018-0118

Article Open Access

Regularity of fuzzy convergence spaces

Lingqiang Li , Qiu Jin and Bingxue Yao

Published/Copyright: December 27, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

(Fuzzy) convergence spaces are extensions of (fuzzy) topological spaces. ⊤-convergence spaces are one of important fuzzy convergence spaces. In this paper, we present an extending dual Fischer diagonal condition, and making use of this we discuss a regularity of ⊤-convergence spaces.

Keywords: Fuzzy set; Fuzzy topology; Fuzzy convergence; Diagonal condition; Regularity

MSC 2010: 54A40; 06D10

1 Introduction

The notion of convergence spaces (refer to [1] for convergence spaces) is introduced by extending the theory of convergence in general topological spaces. For a set X, we denote the power set (resp., the set of filters) on X as P(X) (resp., 𝓕(X)). Then a convergence space is defined as a pair (X, Q), where Q ⊆ 𝓕(X) × X is a binary relation satisfying:

(C1) (ẋ, x) ∈ Q for any x ∈ X, where ẋ = {A ∈ P(X)|x ∈ A} is the principal filter generated by x;

(C2) ∀ 𝓕, 𝓖 ∈ 𝓕(X), 𝓕 ⊆ 𝓖 and (𝓕, x) ∈ Q imply (𝓖, x) ∈ Q.

If (𝓕, x) ∈ Q then we say that 𝓕 converges to x, and denote it as 𝓕 ⟶Qx.

A convergence space (X, Q) is called topological whenever 𝓕 ⟶Qx if and only if 𝓕 converges to x w.r.t some topological space. A convergence space (X, Q) is topological if and only if it satisfies the Fischer diagonal condition.

Let J be any set, Φ : J ⟶ 𝓕(X) and 𝓕 ∈ 𝓕(J), where Φ is called a choice function of filters. Then the Kowalsky compression operator on Φ^⇒(𝓕) ∈ 𝓕(𝓕(X)) is defined as KΦF:=⋃A∈F⋂y∈AΦ(y).

Given a convergence space (X, Q), using Kowalsky compression operator, the Fischer diagonal condition is given as follows.

Let J be any set, ψ : J ⟶ X, Φ : J ⟶ 𝓕(X) such that Φ(j) ⟶Qψ(j) for each j ∈ J. If 𝓕 ∈ 𝓕(X) satisfies ψ^⇒(𝓕) ⟶Qx, then KΦ𝓕 ⟶Qx.
If we take J = X and ψ = id_X in (F) then we get the Kowalsky diagonal condition (K).
A convergence space (X, Q) is called regular if it satisfies the following dual Fischer diagonal condition.
Let J be any set, ψ : J ⟶ X, Φ : J ⟶ 𝓕(X) such that Φ(j) ⟶Qψ(j) for each j ∈ J. If 𝓕 ∈ 𝓕(X) satisfies KΦ𝓕 ⟶Qx, then ψ^⇒(𝓕) ⟶Qx.

For a convergence space generated by a topological space, the convergence space is regular if and only if the corresponding topological space is regular.

In recent years, many kinds of fuzzy convergence spaces, such as stratified L-generalized convergence spaces [2], stratified L-convergence spaces [3], L-ordered convergence spaces [4, 5] and ⊤-convergence spaces [6, 7], were defined and discussed [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In particular, the regularities of fuzzy convergence spaces were discussed by different kinds of extending dual Fischer diagonal conditions [6, 8, 9, 13, 16, 17, 18]. In this paper, we introduce a regularity for ⊤-convergence spaces by an extending dual Fischer diagonal condition.

The contents are arranged as follows. In Section 2, we recall some basic notions. In Section 3, we present the main results : the notion of ⊤-regular ⊤-convergence spaces, and their relationships to regular convergence spaces, and an extension theorem of continuous function. In Section 4, we conclude with a summary.

2 Preliminaries

A commutative quantale is a pair (L, ∗), where L is a complete lattice with respect to a partial order ≤ on it, with the top (resp., bottom) element ⊤ (resp., ⊥), and ∗ is a commutative semigroup operation on L such that a ∗ ⋁_{j ∈ J}b_j = ⋁_{j ∈ J}(a ∗ b_j) for all a ∈ L and {b_j}_{j ∈ J} ⊆ L. (L, ∗) is said to be integral if the top element ⊤ is the unique unit in the sense of ⊤ ∗ a = a for all a ∈ L. (L, ∗) is said to be meet continuous if the underlying lattice (L, ≤) is a meet continuous lattice, that is, the binary meet operation ∧ distributes over directed joins [26]. In this paper, if not otherwise specified, L = (L, ∗) is always assumed to be an integral, commutative, and meet continuous quantale.

Since the binary operation * distributes over arbitrary joins, the function a * (–) : L ⟶ L has a right adjoint a →(–) : L ⟶ L given by a → b = ⋁{c ∈ L : a ∗ c ≤ b}. We collect here some basic properties of the binary operations ∗ and → [27, 28]:

(1) a → b = ⊤ ⇔ a ≤ b; (2) a * b ≤ c ⇔ b ≤ a → c; (3) a∗(a → b) ≤ b; (4) a →(b → c) = (a ∗ b) → c; (5) (⋁_j ∈ Ja_j) → b = ⋀_{j ∈ J}(a_j → b); (6) a →(⋀_{j ∈ J}b_j) = ⋀_{j ∈ J}(a → b_j).

We call a function μ : X → L an L-fuzzy subset in X. We use L^X to denote the set of all L-fuzzy subsets in X. For any A ⊆ X, let ⊤_A denote the characteristic function of A. The operators ⋁, ⋀, ∗ and → on L can be translated onto L^X in a pointwise way. That is, for all μ_t(t ∈ T) ∈ L^X,

(⋁t∈Tμt)(x)=⋁t∈Tμt(x),(⋀t∈Tμt)(x)=⋀t∈Tμt(x),(μ∗ν)(x)=μ(x)∗ν(x),(μ→ν)(x)=μ(x)→ν(x).

Let f : X ⟶ Y be a function. We define f^→ : L^X ⟶ L^Y and f^← : L^Y ⟶ L^X, [27], by f^→(μ)(y) = ⋁_{f(x) = y}μ(x) for μ ∈ L^X and y ∈ Y, and f^←(ν)(x) = ν (f(x)) for ν ∈ L^Y and x ∈ X.

Let μ, ν be L-fuzzy subsets in X. The subsethood degree [29, 30, 31, 32, 33] of μ, ν, denoted as S_X(μ, ν), is defined by SX(μ,ν)=⋀x∈X(μ(x)→ν(x)).

Lemma 2.1

([6, 29, 34, 35, 36, 37, 38]). Let f : X ⟶ Y be a function and μ₁, μ₂ ∈ L^X, λ₁, λ₂ ∈ L^Y. Then

S_X(μ₁, μ₂) ≤ S_Y(f^→ (μ₁), f^→(μ₂)),
S_Y(λ₁, λ₂) ≤ S_X(f^←(λ₁), f^←(λ₂)).

Definition 2.2

([27, 39]). A nonempty subset 𝔽 ⊆ L^X is called a ⊤-filter on the set X whenever:

⋁x∈Xλ(x) = ⊤ for all λ ∈ 𝔽;
λ ∧ μ ∈ 𝔽 for all λ, μ ∈ 𝔽;
if λ ∈ L^X such that⋁μ∈FSX(μ,λ)=⊤,then λ ∈ 𝔽.

It is easily seen that the condition (TF3) implies a weaker condition (TF3′) μ ∈ 𝔽 and μ ≤ λ ⟹ λ ∈ 𝔽.

The set of all ⊤-filters on X is denoted byFL⊤(X).

Definition 2.3

([27]). A nonempty subset 𝔹 ⊆ L^X is called a ⊤-filter base on the setXprovided:

⋁x∈X λ(x) = ⊤ for all λ ∈ 𝔹;
if λ, μ ∈ 𝔹, then⋁ν∈BS_X(ν, λ ∧ μ) = ⊤.

Each ⊤-filter base 𝔹 on X generates a ⊤-filter 𝔽_𝔹defined by 𝔽_𝔹 := {λ ∈ L^X|⋁_{μ ∈ 𝔹}S_X(μ, λ) = ⊤}. And for any λ ∈ L^X, we have the following equality [23]: ⋁μ∈BSX(μ,λ)=⋁μ∈FBSX(μ,λ).

We list some fundamental facts about ⊤-filters in the following proposition.

Proposition 2.4

([6, 27]).

For any x ∈ X, the family [x]_⊤ =: {λ ∈ L^X|λ(x) = ⊤} is a ⊤-filter on X, called the principal ⊤-filter on X generated by x.
For any {𝔽_i}_{i ∈ I} ⊆ FL⊤(X), ⋂_i∈I 𝔽_iis also a ⊤-filter.
Let f : X ⟶ Y be a function. For any 𝔽 ∈ FL⊤(X), 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. For any 𝔾 ∈ FL⊤(Y), the family {f^←(μ)|μ ∈ 𝔾} forms a ⊤-filter base on X if and only if ⋁_x ∈ Xμ(f(x)) = ⊤ holds for all μ ∈ 𝔾, and the ⊤-filter f^⇐(𝔾) (if exists) generated by it is called the inverse image of 𝔾 under f.

Lemma 2.5

Let 𝔽, 𝔾 ∈ FL⊤(X) and 𝔹 be a ⊤-filter base of 𝔽. Then 𝔹 ⊆ 𝔾 implies that 𝔽 ⊆ 𝔾.
Let f : X ⟶ Y be a function and 𝔽 ∈ FL⊤(X). Then λ ∈ f^⇒(𝔽) if and only if f^←(λ) ∈ 𝔽.

Proof

For any λ ∈ 𝔽, we have
⊤=⋁μ∈BSX(μ,λ)≤⋁μ∈GSX(μ,λ),
which means λ ∈ 𝔾, as desired.
Let λ ∈ f^⇒(𝔽). Then
⊤=⋁μ∈FSY(f→(μ),λ)≤⋁μ∈FSX(f←f→(μ),f←(λ))≤⋁μ∈FSX(μ,f←(λ)).
It follows that f^←(λ) ∈ 𝔽. Conversely, let f^←(λ) ∈ 𝔽. Then λ≥ f^→f^←(λ) ∈ f^⇒(𝔽), and so λ ∈ f^⇒(𝔽). □
Let 𝔽 ∈ FL⊤(X), it is easily seen that the set ι(𝔽) = {A ⊆ X|⊤_A ∈ 𝔽} is a filter on X. Conversely, let 𝓕 ∈ 𝓕(X), then the set {⊤_A|A ∈ 𝓕} forms a ⊤-filter base on X and the ⊤-filter generated by it is denoted as ω(𝓕).

Lemma 2.6

Let f : X ⟶ Y, 𝓕 ∈ 𝓕(X), 𝔽 ∈ FL⊤(X) and x ∈ X. Then:

ιω(𝓕) = 𝓕,
ωι(𝔽) ⊆ 𝔽,
ω(ẋ) = [x]_⊤,
ι([x]_⊤) = ẋ,
ι (f^⇒(𝔽)) = f^⇒(ι (𝔽)).

Proof

We prove only (5) and others are easily observed. Indeed,

A∈ι(f⇒(F))⇔f←(⊤A)∈F⇔⊤f←(A)∈F⇔f←(A)∈ι(F)⇔A∈f⇒(ι(F)).

□

Definition 2.7

([6]). A ⊤-convergence space is a pair (X, q), where q ⊆ FL⊤(X) × X is a binary relation satisfying (TC1) ([x]_⊤, x) ∈ q for every x ∈ X; (TC2) if (𝔽, x) ∈ q and 𝔽 ⊆ 𝔾, then (𝔾, x) ∈ q.

If (𝔽, x) ∈ q, then we say that 𝔽 converges to x, and denote it as 𝔽 ⟶qx.

It is easily seen that a ⊤-convergence space is precisely a convergence space when L = {⊥, ⊤}.

For the categorical theory, we refer to the monograph [40].

3 A regularity for ⊤-convergence spaces

In this section, we shall discuss a regularity for ⊤-convergence spaces by an extending dual Fischer diagonal condition.

Let J be any set, ϕ : J ⟶ FL⊤(X) and 𝔽 ∈ FL⊤(J), where ϕ is called a choice function of ⊤-filters. Then the extending Kowalsky compression operator on ϕ^⇒(𝔽) ∈ FL⊤( FL⊤(X)) is defined as

kϕF:=⋃A∈ι(F)⋂y∈Aϕ(y).

We prove that kϕ 𝔽 satisfies (TF1)-(TF3).

(TF1) : Let λ ∈ kϕ 𝔽. Then there exists an A ∈ ι(𝔽) such that for any y ∈ A, λ ∈ ϕ(y). It follows by ϕ(y) ∈ FL⊤(X) that ⋁x∈Xλ(x) = ⊤. Thus the condition (TF1) is satisfied.

(TF2) : Let λ, μ ∈ kϕ 𝔽. Then there exist A, B ∈ ι(𝔽) such that

λ∈⋂y∈Aϕ(y)andμ∈⋂z∈Bϕ(z).

It follows that A ∩ B ∈ ι(𝔽) and λ∧μ∈⋂w∈A∩Bϕ(w), and then λ ∧ μ ∈ kϕ 𝔽. Thus the condition (TF2) is satisfied.

(TF3) : Let λ ∈ L^X satisfy ⋁μ∈kϕFSX(μ,λ)=⊤. Then for any μ ∈ kϕ 𝔽, there exists an A ∈ ι(𝔽) such that for any y ∈ A, μ ∈ ϕ(y). By μ ∈ ϕ(y) we have ⋁ν∈ϕ(y)SX(ν,μ)=⊤. Then it follows that for any y ∈ A,

⊤=⋁μ∈kϕF(SX(μ,λ)∗⋁ν∈ϕ(y)SX(ν,μ))≤⋁ν∈ϕ(y)SX(ν,λ).

That means λ ∈ ϕ(y), and so λ ∈ kϕ 𝔽. Thus the condition (TF3) is satisfied.

Using Kowalsky compression operator, an extension of the (dual) diagonal condition (F) ((DF)) is given as follows:

(TF) Let J be any set, ψ : J ⟶ X, ϕ : J ⟶ FL⊤(X) such that ϕ(j) ⟶qψ(j) for each j ∈ J. If 𝔽 ∈ FL⊤(X) satisfies ψ^⇒(𝔽) ⟶qx, then kϕ 𝔽 ⟶qx.

(TDF) Let J be any set, ψ : J ⟶ X, ϕ : J ⟶ FL⊤(X) such that ϕ(j) ⟶qψ(j) for each j ∈ J. If 𝔽 ∈ FL⊤(X) satisfies kϕ 𝔽 ⟶qx, then ψ^⇒(𝔽) ⟶qx.

If we take J = X and ψ = id_X in (TF) then we get the Kowalsky diagonal condition (TK).

Definition 3.1

A ⊤-convergence space is called ⊤-regular if it satisfies the condition (TDF).

3.1 ⊤-regularity is a good extension of regularity

Let (X, Q) be a convergence space. We define δ(X, Q) = (X, δ (Q)) as

∀F∈FL⊤(X),∀x∈X,F⟶δ(Q)x⇔ι(F)⟶δ(Q)x.

Then it is easily seen that (X, δ(Q)) is a ⊤-convergence space. In this subsection, we shall prove that (X, δ(Q)) is ⊤-regular if and only if (X, Q) is regular. In this sense, we say that ⊤-regularity is a good extension of regularity.

Lemma 3.2

Let f : X ⟶ Y, ϕ : J ⟶ FL⊤(X) and Φ : J ⟶ 𝓕(X). Then for any 𝔽 ∈ FL⊤(J) and 𝓕 ∈ 𝓕(J), we have:

f^⇒(kϕ𝔽) = k(f^⇒ ∘ ϕ)𝔽,
Take Φ₁ = ι ∘ ϕ, then ι(kϕ𝔽) = KΦ₁ι(𝔽),
Take ϕ₁ = ω ∘ Φ, then ι(kϕ₁ω(𝓕)) = KΦ 𝓕,
If σ : J ⟶ FL⊤(X) satisfies σ(j) ⊆ ϕ(j) for any j ∈ J, then kσ𝔽 ⊆ kϕ𝔽.

Proof

For any λ ∈ L^Y, we have
λ∈f⇒(kϕF)⇔f←(λ)∈kϕF⇔∃A∈ι(F),s.tf←(λ)∈⋂y∈Aϕ(y)⇔∃A∈ι(F),s.tλ∈⋂y∈A(f⇒∘ϕ)(y)⇔λ∈k(f⇒∘ϕ)F.
It follows by
A∈ι(kϕF)⇔⊤A∈kϕF⇔∃B∈ι(F),s.t⊤A∈⋂y∈Bϕ(y)⇔∃B∈ι(F),s.tA∈⋂y∈B(ι∘ϕ)(y)=⋂y∈BΦ1(y)⇔A∈KΦ1ι(F).
It follows by
A∈ι(kϕ1ω(F))⇔⊤A∈kϕ1ω(F)⇔∃B∈ι∘ω(F),s.t⊤A∈⋂y∈Bϕ1(y)=⋂y∈B(ω∘Φ)(y)⇔∃B∈F,s.tA∈⋂y∈BΦ(y)⇔A∈KΦF.
It is obvious. □

Theorem 3.3

(X, Q) satisfies (DF) if and only if (X, δ(Q)) satisfies (TDF).

Proof

Let (X, Q) satisfy (DF). Assume that ψ : J ⟶ X, ϕ : J ⟶ FL⊤(X) such that ϕ(j) ⟶δ(Q)ψ(j) for each j ∈ J.

Take Φ = ι ∘ ϕ, it follows by ϕ(j) ⟶δ(Q)ψ(j) that

Φ(j)=ι(ϕ(j))⟶Qψ(j),∀j∈J.

Assume that kϕ𝔽 ⟶δ(Q)x, then by Lemma 3.2(2) we have KΦι(𝔽) = ι(kϕ𝔽) ⟶Qx. It follows by (DF) and Lemma 2.6(5) we get

ι(ψ⇒(F))=ψ⇒(ι(F))⟶Qx,

i.e., ψ^⇒(𝔽) ⟶δ(Q)x. Thus the condition (TDF) is satisfied.

Let (X, δ(Q)) satisfy (TDF). Assume that ψ : J ⟶ X, Φ : J ⟶ 𝓕(X) such that Φ(j) ⟶Qψ(j) for each j ∈ J. Take ϕ = ω ∘ Φ, it follows by Lemma 2.6 (1) that

ι∘ϕ(j)=ι∘ω∘Φ(j)=Φ(j)⟶Qψ(j),

i.e., ϕ(j) ⟶δ(Q)ψ(j) for any j ∈ J.

Let KΦ 𝓕 ⟶Qx. Then by Lemma 3.2 (3) we get ι(kϕω(𝓕)) = KΦ 𝓕 ⟶Qx, i.e., kϕω(𝓕) ⟶δ(Q)x. It follows by (TDF) that ψ^⇒(ω(𝓕)) ⟶δ(Q)x, and then by Lemma 2.6 (5), (1) we have

ψ⇒(F)=ψ⇒((ι∘ω)(F))=ιψ⇒(ω(F))⟶Qx.

Thus the condition (DF) is satisfied. □

3.2 The category of ⊤-regular ⊤-convergence spaces is a reflective subcategory of ⊤-convergence spaces

In this subsection, we shall prove that the category of ⊤-regular convergence spaces is a reflective subcategory of ⊤-convergence spaces.

A function f : X ⟶ Y between two ⊤-convergence spaces (X, q), (Y, p) is called continuous if f^⇒(𝔽) ⟶pf(x) whenever 𝔽 ⟶qx. The category T-CS has as objects all ⊤-convergence spaces and as morphisms the continuous functions.

It is proved in [6] that the category T-CS is topological over SET in the sense of [40]. Indeed, for a given source (X⟶fi (X_i, q_i))_{i ∈ I}, the initial structure, q on X is defined by F⟶qx⇔∀i∈I,fi⇒(F)⟶qifi(x).

Let (X, q) be a ⊤-convergence space, A a subset of X and i_A : A ⟶ X the inclusion function. Then the initial ⊤-convergence structure on A w.r.t. the source i_A : A ⟶ (X, q) is called the substructure of (X, q) on A, denoted by q_A, where

∀x∈A,F∈FL⊤(A),F⟶qAx⇔iA⇒(F)⟶qx.

The pair (X, q_A) is called a subspace of (X, q).

Let X be a nonempty set and let ⊤(X) denote the set of all ⊤-convergence structures on X. If the identity id_X : (X, q) ⟶ (X, p) is continuous then we say q is finer than p or p is coarser than q, and denote p ≤ q.

Proposition 3.4

(⊤(X), ≤) forms a complete lattice.

Proof

For any {q_i}_{i ∈ I} ⊆ ⊤(X), the supremum q of {q_i}_{i ∈ I} exists and is denoted as sup{q_i}. Indeed, sup{q_i} is precisely the initial structures q w.r.t. the source (X⟶idX (X, q_i))_{i ∈ I}, i.e., q = ∩{q_i}_{i ∈ I}. □

In the following, we denote the full subcategory of T-CS consisting of all objects obeying (TDF) as TDF-CS.

Theorem 3.5

The categoryTDF-CSis a topological category overSET.

Proof

We need only check that TDF-CS has initial structure. Assume that (X⟶fi (X_i, q_i))_{i ∈ I} is a source in T-CS such that each (X_i, q_i) ∈ TDF-CS. Let q be the initial structure of the above in TF-CS, that is

F⟶qx⇔∀i∈I,fi⇒(F)⟶qifi(x).

We prove below that (X, q) ∈TDF-CS. Assume that ψ : J ⟶ X, ϕ : J ⟶ FL⊤(X) such that ϕ(j) ⟶qψ(j) for each j ∈ J. It follows that for any i ∈ I, fi⇒(ϕ(j)) ⟶qif_i(ψ(j)). Take ϕ_i = fi⇒ ∘ ϕ and ψ_i = f_i ∘ ψ, then

ϕi(j)⟶qiψi(j),∀j∈J.

Let kϕ𝔽 ⟶qx. Then by Lemma 3.2 (2) we have ∀ i ∈ I,

kϕiF=k(fi⇒∘ϕ)F=fi⇒(kϕF)⟶qifi(x).

By (X, q_i) satisfies (TDF) we have

ψi⇒(F)=(fi∘ψ)⇒(F)=fi⇒ψ⇒(F)⟶qifi(x).

It follows that ψ^⇒(𝔽) ⟶qx. Thus (X, q) satisfies the condition (TDF). □

Let ⊤_DF(X) denote the set of all ⊤-convergence structures on X satisfying (TDF). Then it follows from the above theorem and Proposition 3.4 we get the following corollary.

Corollary 3.6

(⊤_DF(X), ≤) forms a complete lattice.

Theorem 3.7

TDF-CSis a reflective subcategory ofT-CS.

Proof

Let (X, q) ∈T-CS. From Corollary 3.6, the supremum in (⊤(X), ≤) of all s ≤ q with s ∈ ⊤_DF(X), denoted as (X, rq), is also in ⊤_DF(X). Indeed, rq is the finest structure coarser than q satisfying (X, rq) ∈TDF-CS. Hence id_X : (X, q) ⟶ (X, rq) is continuous. Assume that f(X, q) ⟶ (Y, p) is continuous, where (X, p) ∈ TDF-CS. Let s denote the initial structure w.r.t. f : X ⟶ (Y, p). Then (X, s) ∈ TDF-CS and s is the coarsest structure such that f : (X, s) ⟶ (Y, p) is continuous. It follows that s ≤ q and so s ≤ rq. Therefore, f : (X, rq) ⟶ (Y, p) is continuous and thus TDF-CS is reflective in T-CS. □

3.3 Extension of continuous function

In this subsection, based on ⊤-regularity, we shall present an extension theorem of continuous function in the framework of ⊤-convergence space.

Lemma 3.8

Let (A, q_A) be a subspace of a ⊤-convergence space (X, q). If (X, q) fulfils (TK) then (A, q_A) also fulfils this condition.

Proof

Assume that ϕ:A⟶FL⊤(A) satisfies ϕ(y)⟶qAqAy for each y ∈ A. Take ϕ¯:X⟶FL⊤(X) as

ϕ¯(y)=iA⇒(ϕ(y))ify∈Aandϕ¯(y)=[y]⊤ify∉A.

It is easily seen that ϕ¯(y)⟶qy,∀y∈X.

Let F⟶qAqAx, then iA⇒(F)⟶qx. By (X, q) satisfies (TK) we have kϕ¯iA⇒(F)⟶qx. We prove below iA⇒(kϕF)⊇kϕ¯iA⇒(F). Indeed,

λ∈kϕ¯iA⇒(F)⟹∃B∈ι(iA⇒(F))s.t.∀y∈B,λ∈ϕ¯(y)byLemma2.6(5)⟹∃B∈iA⇒(ι(F))s.t.∀y∈B,λ∈ϕ¯(y)⟹A∩B∈ι(F)s.t.∀y∈A∩B,λ∈ϕ¯(y)⟹∃C∈ι(F)s.t.∀y∈C,λ∈iA⇒(ϕ(y))⟹∃C∈ι(F)s.t.∀y∈C,iA←(λ)∈ϕ(y)⟹iA←(λ)∈kϕF⟹λ∈iA⇒(kϕF).

By kϕ¯iA⇒(F)⟶qx we have iA⇒(kϕF)⟶qx, i.e., kϕF⟶qAqAx, as desired. Thus (X, q_A) satisfies the condition (TK). □

Proposition 3.9

Let (X, q) be a ⊤-convergence space satisfying (TK) and (Y, p) be ⊤-regularity. IfAis a nonempty subset ofXsuch that a functionφ : (A, q_A) ⟶ (Y, p) is continuous, thenφhas a continuous extensionφ : (B, q_B) ⟶ (Y, p), where

B={x∈X|CL⊤(x)≠∅,{y|∀F∈CL⊤(x),φ⇒(iA⇐(F))⟶py}≠∅},CL⊤(x)={F∈FL⊤(X)|iA⇐(F)existsandF⟶qx}.

Proof

we prove that A ⊆ B.
For z ∈ A, note that [z]_⊤⟶qz and iA⇐ ([z]_⊤) exists, thus [z]_⊤ ∈ CL⊤(z), which means CL⊤(z) ≠ ∅. Moreover, for any 𝔽 ∈ CL⊤(z), we have iA⇐(𝔽) exists and 𝔽 ⟶qz, then it follows that
iA⇒iA⇐(F)⊇F⟶qz⟹iA⇐(F)⟶qAqAz.
By the continuity of φ we get that φ⇒(iA⇐(F))⟶pφ(z). Thus z ∈ B, and so A ⊆ B.
We extend φ : A ⟶ Y to φ : B ⟶ Y by φ(z) = φ(z) if z ∈ A and φ(z) = y_z, if z ∈ B − A, where y_z is some fixed element in {y|∀F∈CL⊤(z),φ⇒(iA⇐(F))⟶py}. Next, we prove that φ : (B, q_B) ⟶ (Y, p) is continuous. We need to check that for any 𝔾 ∈ FL⊤(B) and any z₀ ∈ B, that G⟶qBz0 implies φ¯⇒(G)⟶pφ¯(z0). We complete it by several steps as follows.

We define a function ϕ_B : B ⟶ FL⊤(B) as ϕ_B(z) = iB⇐(ℍ_z) for any z ∈ B, where ℍ_z ∈ CL⊤(z). Indeed, by iA⇐(ℍ_z) exists and A ⊆ B we get that iB⇐(ℍ_z) exists. Thus ϕ_B is well-defined. Note that (X, q) satisfies (TK), it follows by Lemma 3.8 that (X, q_B) also satisfies (TK). Thus by G⟶qBz0 and
iB⇒iB⇐(Hz)⊇Hz⟶qz⟹ϕB(z)=iB⇐(Hz)⟶qBz,
we get that kϕBG⟶qBz0, i.e., iB⇒(kϕBG)⟶qz0.
iA⇐(kϕ_B𝔾) exists. We need only check that ⋁_z∈A λ(z) = ⊤ for any λ ∈ kϕ_B𝔾. Indeed, it follows by λ ∈ kϕ_B𝔾 that there exists an E ∈ ι(𝔾) such that λ ∈ ϕ_B(e) = iB⇐(ℍ_e) for any e ∈ E. Then
⊤=⋁μ∈HeSB(iB←(μ),λ)≤⋁μ∈HeSA(iA←iB←(μ),iA←(λ))≤⋁μ∈He((⋁z∈Aμ(z))→(⋁z∈Aλ(z))).
Note that ⋁_z∈Aμ(z) = ⊤ since μ ∈ ℍ_e and iA⇐(ℍ_e) exists. It follows that ⋁_z∈Aλ(z) = ⊤, and so iA⇐(kϕ_B𝔾) exists.
iA⇐iB⇒(kϕBG)=iA⇐(kϕBG). It follows by
λ∈iA⇐iB⇒(kϕBG)⇔⋁μ∈kϕBGSA(iA←iB→(μ),λ)=⊤⇔⋁μ∈kϕBGSA(iA←(μ),λ)=⊤⇔λ∈iA⇐(kϕBG).
A combination of (I)-(III) we have iB⇒(kϕ_B𝔾) ⟶qz₀ and iA⇐iB⇒(kϕBG)=iA⇐(kϕBG) exists. It follows that iB⇒(kϕBG)∈CL⊤(z0).
φ⇒iA⇐(kϕBG)=φ⇒iA⇐iB⇒(kϕBG)⟶pφ¯(z0). Indeed, if z₀ ∈ A, then
iA⇒iA⇐iB⇒(kϕBG)⊇iB⇒(kϕBG)⟶qz0,
which means iA⇐iB⇒(kϕBG)⟶qAqAz0, then by the continuity of φ : A ⟶ Y we get φ⇒iA⇐iB⇒(kϕBG)⟶pφ(z₀) = φ(z₀).
If z₀ ∈ B − A, then
φ¯(z0)=yz0∈{y|∀F∈CL⊤(z0),φ⇒(iA⇐(F))⟶py},
and by iB⇒(kϕBG)∈CL⊤(z0), we conclude that φ⇒iA⇐iB⇒(kϕBG)⟶pyz0.
Let ϕ_Y : B ⟶ FL⊤(Y) be the composition of the following three functions
B⟶ϕBFL⊤(B)⟶iA⇐FL⊤(A)⟶φ⇒FL⊤(Y).
Note that for any z∈B,(iA⇐∘ϕB)(z)=iA⇐iB⇐(Hz) exists since λ ∈ ℍ_z implies ⋁_w∈A λ(w) = ⊤. Therefore, ϕ_Y is well-defined. Next, we check that kϕ_Y𝔾 ⟶pφ(z₀) and ϕ_Y(z) ⟶pφ(z) for any z ∈ B.

kϕ_Y𝔾 ⟶pφ(z₀). At first, we prove that iA⇐(kϕBG)⊆k(iA⇐∘ϕB)G. Let λ ∈ iA⇐(kϕ_B𝔾). Then ⋁μ∈kϕBGSA(iA←(μ),λ)=⊤. Note that
μ∈kϕBG⟹∃E∈ι(G)s.t.∀e∈E,μ∈ϕB(e)⟹∃E∈ι(G)s.t.∀e∈E,iA←(μ)∈(iA⇐∘ϕB)(e)⟹iA←(μ)∈k(iA⇐∘ϕB)G.
It follows that
⊤=⋁μ∈kϕBGSA(iA←(μ),λ)≤⋁iA←(μ)∈k(iA⇐∘ϕB)GSA(iA←(μ),λ)≤⋁ν∈k(iA⇐∘ϕB)GSA(ν,λ),
which means λ∈k(iA⇐∘ϕB)G. Thus iA⇐(kϕBG)⊆k(iA⇐∘ϕB)G. Then by Lemma 3.2 (1) we have
φ⇒(iA⇐(kϕBG))⊆φ⇒(k(iA⇐∘ϕB)G)=k(φ⇒∘iA⇐∘ϕB)G=kϕYG,
and by φ^⇒iA⇐(kϕ_B𝔾) ⟶pφ(z₀), it holds that kϕ_Y𝔾 ⟶pφ(z₀).
ϕ_Y(z) ⟶pφ(z) for any z ∈ B. Note that iA⇐(iB⇒(ϕB(z))) exists since
iA⇐(iB⇒(ϕB(z)))=iA⇐(iB⇒(iB⇐(Hz)))=iA⇐(Hz).
Then by iB⇒(ϕB(z))=iB⇒(iB⇐(Hz))⊇Hz⟶qz we obtain iB⇒(ϕ_B(z)) ∈ CL⊤(z).
If z ∈ A, then
iA⇒iA⇐(ϕB(y))=iA⇒iA⇐(iB⇒(iB⇐(Hz)))⊇Hz⟶qz,
which means iA⇐(ϕB(y))⟶qAqAz and so ϕ_Y(y) = φ⇒iA⇐(ϕB(y))⟶pφ(z)=φ¯(z) by the continuity of φ : A ⟶ Y.
If z ∈ B − A, then
φ¯(z)∈{y|∀F∈CL⊤(z),φ⇒(iA⇐(F))⟶py},
and by iB⇒(ϕ_B(z)) ∈ CL⊤(z), we conclude that φ⇒iA⇐iB⇒(ϕB(z))⟶pφ¯(z). Note that
iA⇐(iB⇒(ϕB(z)))=iA⇐(Hz)=iA⇐iB⇐(Hz)=iA⇐(ϕB(z)).
Thus ϕY(z)=φ⇒iA⇐(ϕB(z))⟶pφ¯(z).
It follows from (i), (ii) and that (Y, p) satisfies (TDF) we get that φ¯⇒(G)⟶pφ¯(z0) for any G⟶qBz0. This means that φ : (B, q_B) ⟶ (Y, p) is continuous. □
A subset B of a ⊤-convergence space (X, q) is said to be dense [6] if for each x ∈ X, there exists a ⊤-filter 𝔽 such that iB⇐(𝔽) exists and 𝔽 converges to x. (X, q) is called ⊤-Hausdorff if for each ⊤-filter 𝔽, there exists at most one x ∈ X such that 𝔽 converges to x.

Theorem 3.10

(continuous extension theorem). Let (X, q) be a ⊤-convergence space satisfying Kowalsky ⊤-diagonal condition (TK), and let (Y, p) be regular and ⊤-Hausdorff. Then for each dense subsetAin (X, q), a continuous functionφ : (A, q_A) ⟶ (Y, p) has a unique continuous extensionφ : (X, q) ⟶ (Y, p) if and only if{y|∀F∈CL⊤(x),φ⇒(iA⇐(F))⟶py}≠∅for anyx ∈ X.

Proof

Sufficiency. For any x ∈ X, since A is dense in (X, q) then CL⊤(x) ≠ ∅, it follows by {y∣∀ 𝔽 ∈ CL⊤(x),φ⇒(iA⇐(F))⟶py}≠∅ and we have that

{x∈X|CL⊤(x)≠∅,{y|∀F∈CL⊤(x),φ⇒(iA⇐(F))⟶py}≠∅}=X.

From Proposition 3.9, we conclude that there exists a continuous extension of φ, defined as φ : X ⟶ Y : ∀ x ∈ X, φ(x) = φ(x) if x ∈ A and φ(x) = y_x, if x ∈ X − A, where y_x is some fixed element in {y∣∀ 𝔽 ∈ CL⊤(x),φ⇒(iA⇐(F))⟶py}. Note that the set {y|∀F∈CL⊤(x),φ⇒(iA⇐(F))⟶py} has only one element since (Y, p) is ⊤-Hausdorff. This means that φ is defined uniquely.

Necessity. Assume that φ has a continuous extension φ : (X, q) ⟶ (Y, p). Then we have that φ¯⇒(F)⟶pφ(x) for any 𝔽 ⟶qx. Next we check that φ¯⇒(F)⊆φ⇒(iA⇐(F)) whenever iA⇐(𝔽) exists. Indeed, for any λ ∈ 𝔽, it is easily seen that φ→(iA←(λ))≤φ¯→(λ) and so φ¯→(λ)∈φ⇒(iA⇐(F)). It follows by Lemma 2.5 (1) that φ¯⇒(F)⊆φ⇒(iA⇐(F)).

For any 𝔽 ∈ CL⊤(x), which means 𝔽 ⟶qx and iA⇐(𝔽) exists. From the above statement we observe easily that φ⇒(iA⇐(F))⟶pφ¯(x). Therefore, {y|∀F∈CL⊤(x),φ⇒(iA⇐(F))⟶py}≠∅. □

4 Conclusions

In this paper, we defined a notion of ⊤-regularity for ⊤-convergence spaces with the use of an extending dual Fischer diagonal condition, which is based on extending Kowalsky compression operator. It is proved that ⊤-regularity is a good extension of regularity, and the category of ⊤-regular ⊤-convergence space is a reflective category of ⊤-convergence spaces. In addition, based on ⊤-regularity, we explored an extension theorem of continuous function.

Acknowledgement

The authors thank the reviewers and the editor for their valuable comments and suggestions. This work is supported by National Natural Science Foundation of China (11501278, 11801248, 11471152) and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD03).

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Received: 2018-05-15

Accepted: 2018-10-25

Published Online: 2018-12-27

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Articles in the same Issue

https://doi.org/10.1515/math-2018-0118

Keywords for this article

Fuzzy set; Fuzzy topology; Fuzzy convergence; Diagonal condition; Regularity

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