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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Regularity of fuzzy convergence spaces

Lingqiang Li / Qiu Jin
  • College of Mathematics and Sciences, Liaocheng University, Liaocheng, China
  • School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China
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  • De Gruyter OnlineGoogle Scholar
/ Bingxue Yao
Published Online: 2018-12-27 | DOI: https://doi.org/10.1515/math-2018-0118

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) (, x) ∈ Q for any xX, where = {AP(X)|xA} 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 Q x.

A convergence space (X, Q) is called topological whenever 𝓕 Q x 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:=AFyAΦ(y).

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

  • (F)

    Let J be any set, ψ : JX, Φ : J ⟶ 𝓕(X) such that Φ(j) Q ψ(j) for each jJ. If 𝓕 ∈ 𝓕(X) satisfies ψ(𝓕) Q x, then 𝓕 Q x.

    If we take J = X and ψ = idX 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.

  • (DF)

    Let J be any set, ψ : JX, Φ : J ⟶ 𝓕(X) such that Φ(j) Q ψ(j) for each jJ. If 𝓕 ∈ 𝓕(X) satisfies 𝓕 Q x, then ψ(𝓕) Q x.

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 ∗ ⋁jJ bj = ⋁jJ(abj) for all aL and {bj}jJL. (L, ∗) is said to be integral if the top element ⊤ is the unique unit in the sense of ⊤ ∗ a = a for all aL. (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 * (–) : LL has a right adjoint a →(–) : LL given by ab = ⋁{cL : acb}. We collect here some basic properties of the binary operations ∗ and → [27, 28]:

(1) ab = ⊤ ⇔ ab; (2) a * bcbac; (3) a∗(ab) ≤ b; (4) a →(bc) = (ab) → c; (5) (⋁jJaj) → b = ⋀jJ(ajb); (6) a →(⋀jJbj) = ⋀jJ(abj).

We call a function μ : XL an L-fuzzy subset in X. We use LX to denote the set of all L-fuzzy subsets in X. For any AX, let ⊤A denote the characteristic function of A. The operators ⋁, ⋀, ∗ and → on L can be translated onto LX in a pointwise way. That is, for all μt(tT) ∈ LX, (tTμt)(x)=tTμt(x),(tTμt)(x)=tTμt(x),(μν)(x)=μ(x)ν(x),(μν)(x)=μ(x)ν(x).

Let f : XY be a function. We define f : LXLY and f : LYLX, [27], by f(μ)(y) = ⋁f(x) = yμ(x) for μLX and yY, and f(ν)(x) = ν (f(x)) for νLY and xX.

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

Lemma 2.1

([6, 29, 34, 35, 36, 37, 38]). Let f : XY be a function and μ1, μ2LX, λ1, λ2LY. Then

  1. SX(μ1, μ2) ≤ SY(f (μ1), f(μ2)),

  2. SY1, λ2) ≤ SX(f1), f2)).

Definition 2.2

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

  • (TF1)

    xXλ(x) = ⊤ for all λ ∈ 𝔽;

  • (TF2)

    λ ∧ μ ∈ 𝔽 for all λ, μ ∈ 𝔽;

  • (TF3)

    if λLX such that μFSX(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 by FL(X).

Definition 2.3

([27]). A nonempty subset 𝔹 ⊆ LX is called a ⊤-filter base on the set X provided:

  • (TB1)

    xX λ(x) = ⊤ for all λ ∈ 𝔹;

  • (TB2)

    if λ, μ ∈ 𝔹, then νB SX(ν, λ ∧ μ) = ⊤.

Each ⊤-filter base 𝔹 on X generates a ⊤-filter 𝔽𝔹 defined by 𝔽𝔹 := {λ ∈ LX|⋁μ ∈ 𝔹SX(μ, λ) = ⊤}. And for any λLX, we have the following equality [23]: μBSX(μ,λ)=μFBSX(μ,λ).

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

Proposition 2.4

([6, 27]).

  1. For any xX, the family [x] =: {λ ∈ LX|λ(x) = ⊤} is a ⊤-filter on X, called the principal ⊤-filter on X generated by x.

  2. For any {𝔽i}iIFL(X), ⋂iI 𝔽i is also a ⊤-filter.

  3. Let f : XY 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 ifxXμ(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

  1. Let 𝔽, 𝔾 ∈ FL(X) and 𝔹 be a ⊤-filter base of 𝔽. Then 𝔹 ⊆ 𝔾 implies that 𝔽 ⊆ 𝔾.

  2. Let f : XY be a function and 𝔽 ∈ FL(X). Then λf(𝔽) if and only if f(λ) ∈ 𝔽.

Proof

  1. For any λ ∈ 𝔽, we have =μBSX(μ,λ)μGSX(μ,λ),

    which means λ ∈ 𝔾, as desired.

  2. Let λ ∈ f(𝔽). Then =μFSY(f(μ),λ)μFSX(ff(μ),f(λ))μFSX(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) = {AX|⊤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 : XY, 𝓕 ∈ 𝓕(X), 𝔽 ∈ FL(X) and xX. Then:

  1. ιω(𝓕) = 𝓕,

  2. ωι(𝔽) ⊆ 𝔽,

  3. ω() = [x],

  4. ι([x]) = ,

  5. ι (f(𝔽)) = f(ι (𝔽)).

Proof

We prove only (5) and others are easily observed. Indeed, Aι(f(F))f(A)Ff(A)Ff(A)ι(F)Af(ι(F)).

Definition 2.7

([6]). A ⊤-convergence space is a pair (X, q), where qFL(X) × X is a binary relation satisfying (TC1) ([x], x) ∈ q for every xX; (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 q x.

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 : JFL(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:=Aι(F)yAϕ(y).

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

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

(TF2) : Let λ, μ 𝔽. Then there exist A, Bι(𝔽) such that λyAϕ(y)andμzBϕ(z).

It follows that ABι(𝔽) and λμwABϕ(w), and then λ ∧ μ 𝔽. Thus the condition (TF2) is satisfied.

(TF3) : Let λ ∈ LX satisfy μkϕFSX(μ,λ)=. Then for any μ 𝔽, there exists an Aι(𝔽) such that for any yA, μϕ(y). By μϕ(y) we have νϕ(y)SX(ν,μ)=. Then it follows that for any yA, =μkϕF(SX(μ,λ)νϕ(y)SX(ν,μ))νϕ(y)SX(ν,λ).

That means λ ∈ ϕ(y), and so λ ∈ 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 : JX, ϕ : JFL(X) such that ϕ(j) qψ(j) for each jJ. If 𝔽 ∈ FL(X) satisfies ψ(𝔽) qx, then kϕ 𝔽 q x.

(TDF) Let J be any set, ψ : JX, ϕ : JFL(X) such that ϕ(j) qψ(j) for each jJ. If 𝔽 ∈ FL(X) satisfies kϕ 𝔽 q x, then ψ(𝔽) qx.

If we take J = X and ψ = idX 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 FFL(X),xX,Fδ(Q)xι(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 : XY, ϕ : JFL(X) and Φ : J ⟶ 𝓕(X). Then for any 𝔽 ∈ FL(J) and 𝓕 ∈ 𝓕(J), we have:

  1. f(𝔽) = k(fϕ)𝔽,

  2. Take Φ1 = ιϕ, then ι(𝔽) = 1ι(𝔽),

  3. Take ϕ1 = ωΦ, then ι(1ω(𝓕)) = 𝓕,

  4. If σ : JFL(X) satisfies σ(j) ⊆ ϕ(j) for any jJ, then kσ𝔽 ⊆ 𝔽.

Proof

  1. For any λ ∈ LY, we have λf(kϕF)f(λ)kϕFAι(F),s.tf(λ)yAϕ(y)Aι(F),s.tλyA(fϕ)(y)λk(fϕ)F.

  2. It follows by Aι(kϕF)AkϕFBι(F),s.tAyBϕ(y)Bι(F),s.tAyB(ιϕ)(y)=yBΦ1(y)AKΦ1ι(F).

  3. It follows by Aι(kϕ1ω(F))Akϕ1ω(F)Bιω(F),s.tAyBϕ1(y)=yB(ωΦ)(y)BF,s.tAyBΦ(y)AKΦF.

  4. 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 : JX, ϕ : JFL(X) such that ϕ(j) δ(Q)ψ(j) for each jJ.

Take Φ = ιϕ, it follows by ϕ(j) δ(Q)ψ(j) that Φ(j)=ι(ϕ(j))Qψ(j),jJ.

Assume that kϕ𝔽 δ(Q) x, then by Lemma 3.2(2) we have KΦι(𝔽) = ι(𝔽) Q x. 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 : JX, Φ : J ⟶ 𝓕(X) such that Φ(j) Qψ(j) for each jJ. Take ϕ = ωΦ, it follows by Lemma 2.6 (1) that ιϕ(j)=ιωΦ(j)=Φ(j)Qψ(j),

i.e., ϕ(j) δ(Q) ψ(j) for any jJ.

Let 𝓕 Q x. Then by Lemma 3.2 (3) we get ι(kϕω(𝓕)) = 𝓕 Q x, 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((ιω)(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 : XY between two ⊤-convergence spaces (X, q), (Y, p) is called continuous if f(𝔽) p f(x) whenever 𝔽 q x. 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 (Xi, qi))iI, the initial structure, q on X is defined by FqxiI,fi(F)qifi(x).

Let (X, q) be a ⊤-convergence space, A a subset of X and iA : AX the inclusion function. Then the initial ⊤-convergence structure on A w.r.t. the source iA : A ⟶ (X, q) is called the substructure of (X, q) on A, denoted by qA, where xA,FFL(A),FqAxiA(F)qx.

The pair (X, qA) 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 idX : (X, q) ⟶ (X, p) is continuous then we say q is finer than p or p is coarser than q, and denote pq.

Proposition 3.4

(⊤(X), ≤) forms a complete lattice.

Proof

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

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

Theorem 3.5

The category TDF-CS is a topological category over SET.

Proof

We need only check that TDF-CS has initial structure. Assume that (X fi (Xi, qi))iI is a source in T-CS such that each (Xi, qi) ∈ TDF-CS. Let q be the initial structure of the above in TF-CS, that is FqxiI,fi(F)qifi(x).

We prove below that (X, q) ∈TDF-CS. Assume that ψ : JX, ϕ : JFL(X) such that ϕ(j) qψ(j) for each jJ. It follows that for any iI, fi(ϕ(j)) qifi(ψ(j)). Take ϕi = fiϕ and ψi = fiψ, then ϕi(j)qiψi(j),jJ.

Let 𝔽 q x. Then by Lemma 3.2 (2) we have ∀ iI, kϕiF=k(fiϕ)F=fi(kϕF)qifi(x).

By (X, qi) satisfies (TDF) we have ψi(F)=(fiψ)(F)=fiψ(F)qifi(x).

It follows that ψ(𝔽) q x. 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-CS is a reflective subcategory of T-CS.

Proof

Let (X, q) ∈T-CS. From Corollary 3.6, the supremum in (⊤(X), ≤) of all sq 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 idX : (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 sq and so srq. 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, qA) be a subspace of a ⊤-convergence space (X, q). If (X, q) fulfils (TK) then (A, qA) also fulfils this condition.

Proof

Assume that ϕ:AFL(A) satisfies ϕ(y)qAqAy for each yA. Take ϕ¯:XFL(X) as

ϕ¯(y)=iA(ϕ(y))ifyAandϕ¯(y)=[y]ifyA.

It is easily seen that ϕ¯(y)qy,yX.

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

λkϕ¯iA(F)Bι(iA(F))s.t.yB,λϕ¯(y)byLemma2.6(5)BiA(ι(F))s.t.yB,λϕ¯(y)ABι(F)s.t.yAB,λϕ¯(y)Cι(F)s.t.yC,λiA(ϕ(y))Cι(F)s.t.yC,iA(λ)ϕ(y)iA(λ)kϕFλiA(kϕF).

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

Proposition 3.9

Let (X, q) be a ⊤-convergence space satisfying (TK) and (Y, p) be ⊤-regularity. If A is a nonempty subset of X such that a function φ : (A, qA) ⟶ (Y, p) is continuous, then φ has a continuous extension φ : (B, qB) ⟶ (Y, p), where

B={xX|CL(x),{y|FCL(x),φ(iA(F))py}},CL(x)={FFL(X)|iA(F)existsandFqx}.

Proof

  1. we prove that AB.

    For zA, note that [z] q z 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 q z, then it follows that

    iAiA(F)FqziA(F)qAqAz.

    By the continuity of φ we get that φ(iA(F))pφ(z). Thus zB, and so AB.

  2. We extend φ : AY to φ : BY by φ(z) = φ(z) if zA and φ(z) = yz, if zBA, where yz is some fixed element in {y|FCL(z),φ(iA(F))py}. Next, we prove that φ : (B, qB) ⟶ (Y, p) is continuous. We need to check that for any 𝔾 ∈ FL(B) and any z0B, that GqBz0 implies φ¯(G)pφ¯(z0). We complete it by several steps as follows.

  1. We define a function ϕB : BFL(B) as ϕB(z) = iB(ℍz) for any zB, where ℍzCL(z). Indeed, by iA(ℍz) exists and AB we get that iB(ℍz) exists. Thus ϕB is well-defined. Note that (X, q) satisfies (TK), it follows by Lemma 3.8 that (X, qB) also satisfies (TK). Thus by GqBz0 and

    iBiB(Hz)HzqzϕB(z)=iB(Hz)qBz,

    we get that kϕBGqBz0, i.e., iB(kϕBG)qz0.

  2. iA(B𝔾) exists. We need only check that ⋁zA λ(z) = ⊤ for any λ ∈ B𝔾. Indeed, it follows by λ ∈ B𝔾 that there exists an Eι(𝔾) such that λ ∈ ϕB(e) = iB(ℍe) for any eE. Then

    =μHeSB(iB(μ),λ)μHeSA(iAiB(μ),iA(λ))μHe((zAμ(z))(zAλ(z))).

    Note that ⋁zAμ(z) = ⊤ since μ ∈ ℍe and iA(ℍe) exists. It follows that ⋁zAλ(z) = ⊤, and so iA(B𝔾) exists.

  3. iAiB(kϕBG)=iA(kϕBG). It follows by

    λiAiB(kϕBG)μkϕBGSA(iAiB(μ),λ)=μkϕBGSA(iA(μ),λ)=λiA(kϕBG).

    A combination of (I)-(III) we have iB(B𝔾) q z0 and iAiB(kϕBG)=iA(kϕBG) exists. It follows that iB(kϕBG)CL(z0).

  4. φiA(kϕBG)=φiAiB(kϕBG)pφ¯(z0). Indeed, if z0A, then

    iAiAiB(kϕBG)iB(kϕBG)qz0,

    which means iAiB(kϕBG)qAqAz0, then by the continuity of φ : AY we get φiAiB(kϕBG)p φ(z0) = φ(z0).

    If z0BA, then

    φ¯(z0)=yz0{y|FCL(z0),φ(iA(F))py},

    and by iB(kϕBG)CL(z0), we conclude that φiAiB(kϕBG)pyz0.

  5. Let ϕY : BFL(Y) be the composition of the following three functions

    BϕBFL(B)iAFL(A)φFL(Y).

    Note that for any zB,(iAϕB)(z)=iAiB(Hz) exists since λ ∈ ℍz implies ⋁wA λ(w) = ⊤. Therefore, ϕY is well-defined. Next, we check that Y𝔾 p φ(z0) and ϕY(z) p φ(z) for any zB.

  1. Y𝔾 p φ(z0). At first, we prove that iA(kϕBG)k(iAϕB)G. Let λ ∈ iA(B𝔾). Then μkϕBGSA(iA(μ),λ)=. Note that

    μkϕBGEι(G)s.t.eE,μϕB(e)Eι(G)s.t.eE,iA(μ)(iAϕB)(e)iA(μ)k(iAϕB)G.

    It follows that

    =μkϕBGSA(iA(μ),λ)iA(μ)k(iAϕB)GSA(iA(μ),λ)νk(iAϕB)GSA(ν,λ),

    which means λk(iAϕB)G. Thus iA(kϕBG)k(iAϕB)G. Then by Lemma 3.2 (1) we have

    φ(iA(kϕBG))φ(k(iAϕB)G)=k(φiAϕB)G=kϕYG,

    and by φ iA(B𝔾) p φ(z0), it holds that Y𝔾 p φ(z0).

  2. ϕY(z) p φ(z) for any zB. 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))Hzqz we obtain iB(ϕB(z)) ∈ CL(z).

    If zA, then

    iAiA(ϕB(y))=iAiA(iB(iB(Hz)))Hzqz,

    which means iA(ϕB(y))qAqAz and so ϕY(y) = φiA(ϕB(y))pφ(z)=φ¯(z) by the continuity of φ : AY.

    If zBA, then

    φ¯(z){y|FCL(z),φ(iA(F))py},

    and by iB(ϕB(z)) ∈ CL(z), we conclude that φiAiB(ϕB(z))pφ¯(z). Note that

    iA(iB(ϕB(z)))=iA(Hz)=iAiB(Hz)=iA(ϕB(z)).

    Thus ϕY(z)=φiA(ϕB(z))pφ¯(z).

    It follows from (i), (ii) and that (Y, p) satisfies (TDF) we get that φ¯(G)pφ¯(z0) for any GqBz0. This means that φ : (B, qB) ⟶ (Y, p) is continuous. □

    A subset B of a ⊤-convergence space (X, q) is said to be dense [6] if for each xX, 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 xX 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 subset A in (X, q), a continuous function φ : (A, qA) ⟶ (Y, p) has a unique continuous extension φ : (X, q) ⟶ (Y, p) if and only if {y|FCL(x),φ(iA(F))py} for any xX.

Proof

Sufficiency. For any xX, 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

{xX|CL(x),{y|FCL(x),φ(iA(F))py}}=X.

From Proposition 3.9, we conclude that there exists a continuous extension of φ, defined as φ : XY : ∀ xX, φ(x) = φ(x) if xA and φ(x) = yx, if xXA, where yx is some fixed element in {y∣∀ 𝔽 ∈ CL(x),φ(iA(F))py}. Note that the set {y|FCL(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)p φ(x) for any 𝔽 q x. 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 q x and iA(𝔽) exists. From the above statement we observe easily that φ(iA(F))pφ¯(x). Therefore, {y|FCL(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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About the article

Received: 2018-05-15

Accepted: 2018-10-25

Published Online: 2018-12-27


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1455–1465, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0118.

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© 2018 Liu et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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