On Fuzzy C-Paracompact Topological Spaces

Abstract

The aim of this paper is to study fuzzy extensions of some covering properties defined by A. V. Arhangel’skii and studied by other authors. Indeed, in 2016, A. V. Arhangel’skii defined other paracompact-type properties: C-paracompactness and C₂-paracompactness. Later, M. M. Saeed, L. Kalantan and H. Alzumi investigated these two properties. In this paper, we define fuzzy extensions of these notions and obtain results about them, and in particular, prove that these are good extensions of those defined by Arhangel’skii.

1. Introduction

In 2016, A. V. Arhangel’skii defined other paracompact-type properties: C-paracompactness and C₂-paracompactness. A topological space X is called C-paracompact if there is a paracompact space Y and a bijective function f: X → Y such that the restriction $f_{|A}:A\to f\left(A\right)$ is a homeomorphism for each compact subspace A ⊂ X. A topological space X is called C₂—paracompact if there is a Hausdorff paracompact space Y and a bijective function f: X → Y f such that the restriction $f_{|A}:A\to f\left(A\right)$ is a homeomorphism for each compact subspace A ⊂ X. Later, M. M. Saeed, L. Kalantan and H. Alzumi [1] investigated these two properties and gave some examples that illustrate relations between them. In this paper, we define fuzzy extensions of these notions and obtain results about them.

2. Definitions

First, we give some previous definitions:

Definition 1.

Let (X,τ) be a fuzzy topological space. We will say that (X,τ) is fuzzy C-paracompact if there exists a fuzzy paracompact space (Y,ζ) (in various senses, which is will specify) and a bijection map f: X → Y such that restriction$f_{|K}:K\to f\left(K\right)$is a fuzzy homeomorphism for each K ⊂ X such that its characteristic map χ_K is a Lowen’s fuzzy compact subset.

Definition 2.

Let (X,τ) be a fuzzy topological space. We will say that (X,τ) is fuzzy C₂-paracompact if there exists a fuzzy paracompact (in various senses, which is will specify in next definitions) Hausdorff space (Y,ζ) and a bijection map f: X → Y, such that restriction$f_{|K}:K\to f\left(K\right)$is a fuzzy homeomorphism for each K ⊂ X such that its characteristic map χ_K is a Lowen’s fuzzy compact subset.

Remark 1.

The kinds of fuzzy paracompact, fuzzy compact and fuzzy Hausdorff spaces cited in the above definitions should be good extensions of paracompactness, compactness and Hausdorff topological spaces. We list these definitions:

Definition 3

([2]). Let r ∊ (0, 1], µ be a set in a fuzzy topological space (X,τ). We say that µ is r-paracompact (resp. r*-paracompact) if for each r-open Q-cover of µ there exists an open refinement of it which is both locally finite (resp. ⋆-locally finite) in μ and a r-Q-cover of μ. Additionally, µ is called S-paracompact (resp. S *-paracompact) if for every r ∊ (0, 1], µ is r-paracompact (resp. r*-paracompact). We say that (X,τ) is r-paracompact (resp. r*-paracompact, S-paracompact, S*-paracompact) if set X verifies this property.

Definition 4

([3]). Let µ be a fuzzy set in a fuzzy topological space (X,τ). We say that μ is fuzzy paracompact (resp. r*-paracompact) if for each open L-cover $𝒰$ of µ and for each r ∊ (0, 1], there exists an open refinement $𝒱$ of $𝒰$ which is both locally finite (resp. ⋆-locally finite) in µ and L-cover of µ − r. We say that a fuzzy topological space (X,τ) is fuzzy paracompact (resp. ⋆-fuzzy paracompact) if each constant fuzzy set in X is fuzzy paracompact (resp. ⋆-fuzzy paracompact).

Definition 5

([4]). A fuzzy topological space (X,τ) is called fuzzy paracompact if for each $𝒰$ ⊂ τ and for each r ∊ (0, 1] such that sup{µ|µ ∊ $𝒰$} ≥ r, and for all ε (0 < ε ≤ r), there exists a locally finite open refinement $𝒱$ of $𝒰$ such that sup{µ|µ ∊ $𝒱$} ≥ r − ε.

Definition 6

([5]). A fuzzy set μ in a fuzzy topological space (X,τ) is called fuzzy compact (in the Lowen’s sense) if for all family $𝒰$ ⊂ τ such that sup{υ|υ ∊ $𝒰$} ≥ µ and for all ε>0 there exists a finite subfamily $𝒰$₀ ⊂ $𝒰$ such that sup{υ|υ ∊ $𝒰$₀} ≥ µ − ε. The fuzzy topological space (X,τ) is fuzzy compact if each constant fuzzy set in (X,τ) is fuzzy compact.

Definition 7

([6]). A fuzzy topological space (X,τ) is said to be fuzzy Hausdorff if for any two distinct fuzzy points p,q ∊ X, there are disjoint $𝒰$,$𝒱$ ∊ τ with p ∊ $𝒰$ and q ∊ $𝒱$.

3. Results

Lemma 1.

Let (X,T) be a topological space and A be a subset of X. Then, A is compact in (X,T) if and only if its characteristic map χ_A is a Lowen’s fuzzy compact subset in (X,ω(T)).

Proof. 

(⇐) For each $𝒰$ ⊂ T, such that $A\subset \underset{𝒰\in U}{{\displaystyle \cup }}U$ is sup{χ_U|U ∊ $𝒰$} ≥ χ_A.

For each ε ∊ (0, 1), from the hypothesis there exists a finite subfamily $𝒰$₀ ⊂ $𝒰$ such that sup{χ_U|U ∊ $𝒰$} ≥ χ_A − ε. Then, $𝒰$₀ is a finite subcovering of $𝒰$.

(⇒) Let $ℱ$ ⊂ ω(T) such that sup{µ ∊ $ℱ$} ≥ χ_A. For each ε > 0 and for each µ ∊ $ℱ$ if µ^ε = µ + ε, and $ℒ$=(µ^ε) = {(x,r)|µ^ε(x) > r} is open in X × R. Additionally, $\underset{\mu \in ℱ}{{\displaystyle \cup }}ℒ\left({\mu }^{\epsilon }\right)$ ⊃ A × I (which is compact), because for each (x,r) ∊ A × I (where ε < r), is sup{µ(x)|µ ∊ $ℱ$} = 1 ≥ r > ε, then, there exists µ₀ ∊ $ℱ$ such that ε < r ≤ µ₀(x).

Thus, µ₀(x) + ε > r > ε, then (x,r) ∊ $ℒ$(µ^ε).

Finally, there exists a finite subfamily $ℱ$₀ ⊂ $ℱ$ such that $\underset{\mu \in ℱ}{{\displaystyle \cup }}ℒ\left({\mu }^{\epsilon }\right)\supset A\times I$ and sup{µ|µ ∊ $ℱ$₀} ≥ χ_A − ε because for each (a,1) ∊ A × I there is µ ∊ $ℱ$₀ such that (a,1) ∊ $ℒ$(µ^ε), then µ₀(a) + ε > 1, and sup{µ|µ ∊ $ℱ$₀} ≥ χ_A − ε. □

Proposition 1.

Let (X,τ) be a fuzzy topological space. Then, (X,τ) is fuzzy C-paracompact if and only if$\left(X,\iota \left(\tau \right)\right)$is C-paracompact, i.e., fuzzy C-paracompactness is a good extension of C-paracompactness.

Proof. 

(X,τ) is fuzzy C-paracompact, i.e., there exists a fuzzy paracompact space (Y,ζ) (in the sense of some good extension of paracompactness [2,3,4,7]) and a bijection map f: X → Y such that restriction $f_{|K}:K\to f\left(K\right)$ is a fuzzy homeomorphism for each K ⊂ X such that its characteristic map χ_K is a Lowen’s fuzzy compact subset [5]. That is, there is a paracompact space $\left(Y,\iota \left(\varsigma \right)\right)$ and a bijection map f: X → Y such that the restriction $f_{|K}:K\to f\left(K\right)$ is a homeomorphism for each compact subspace K ⊂ X, i.e., $\left(X,\iota \left(\tau \right)\right)$ is C-paracompact. □

Corollary 1.

Fuzzy C₂-paracompactness is a good extension of C₂-paracompactness.

Proposition 2.

Every fuzzy Hausdorff topological space (in the Srivastava, Lal and Srivastava or in the Wagner and McLean sense) which is fuzzy locally compact (in the Kudri and Wagner sense) is fuzzy C₂-paracompact.

Proof. 

It follows from ([6], Prop.3.2), ([8], Prop.3.1), ([9], Th.3.3) and above Corollary. (For undefined concepts and previous results see [10]). □