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Soft ω*-Paracompactness in Soft Topological Spaces
International Journal of Fuzzy Logic and Intelligent Systems 2021;21(1):57-65
Published online March 25, 2021
© 2021 Korean Institute of Intelligent Systems.

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
Correspondence to: Samer Al Ghour (algore@just.edu.jo)
Received October 10, 2020; Revised January 6, 2021; Accepted January 19, 2021.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this study, we introduce a new concept in soft topological spaces, namely, soft ω*-paracompactness, and we provide characterizations thereof. Its connection with other related concepts is also studied. In particular, we show that soft ω*-paracompactness and soft paracompactness are independent of each other. In addition, we study the soft ω*-paracompactness of the soft topological space generated by an indexed family of ω*-paracompact topological spaces.
Keywords : ω-open sets, ω*-paracompact, Soft paracompact, Generated soft topology
1. Introduction

Throughout this paper, we use the notions and terminology in [1] and [2]; moreover, TS and STS stand for “topological space” and “soft topological space,” respectively. Recently, classical methods have been applied to several problems in various fields, such as engineering, social sciences, and medical sciences. Soft sets were defined by Molodtsov [3], and they have found numerous applications. Let X be a universal set and A be a set of parameters. A soft set over X relative to A is a function G : A → ℘(X). The family of all soft sets over X relative to A is denoted by SS(X, A). If GSS (X, A) is such that G(a) is a countable set for all aX, then G is called a countable soft set. The family of all members of SS(X, A) that are countable is denoted by CSS (X, A). We denote the null soft set and the absolute soft set as 0A and 1A, respectively. Soft topological spaces were defined in [4] as follows: A triplet (X, τ, A), where τSS (X, A), is called an STS if 0A and 1Aτ, and τ is closed under finite soft intersections and arbitrary soft unions. For an STS (X, τ, A), the members of τ are called soft open sets, and their soft complements are called soft closed sets. Several topological concepts have been extended to the context of STSs [1, 2, 528]. The concept of ω-open set was introduced in [29] as a weaker form of an open set as follows: Let (X, ℑ) be a TS and let UX; then, U is called ω-open if for every xU, there is an open set V and a countable subset CX such that xVCU.

The complement of an ω-open set in (X, ℑ) is called an ω-closed set. The family of all ω-open sets in (X, ℑ) is denoted by ℑω. It is known that ℑω forms a topology on X that contains ℑ. The study of ω-open sets remains an area that attracts considerable attention [3035]. Using ω-open sets, the concept of ω*-paracompactness was defined and studied in ordinary TSs [36]. In [2], the concept of ω-openness was extended to include STSs as follows: Let (X, τ, A) be an STS, and let GSS(X, A); then, G is called a soft ω-open set if for all ax ∊̃ H, there exist Fτ and HCSS(X, A) such that ax ∊̃FH ⊂̃G. The collection of all soft ω-open sets in (X, τ, A) is denoted by τω. It was proved in [2] that (X, τω, A) is an STS with ττω.

In this study, using soft ω-open sets, we introduce a new concept in STSs, namely, ω*-paracompactness, which extends ω*-paracompactness in ordinary topological spaces. As symmetries, we provide characterizations of this concept. We also study the connection of soft ω*-paracompactness with other related topological concepts. In particular, we show that soft ω*-paracompactness and soft paracompactness are independent of each other. Moreover, we study the soft ω*-paracompactness of the STS generated by an indexed family of ω*-paracompact TSs.

2. Preliminaries

Herein, we recall several related definitions and results.

Definition 2.1

Let (X, ℑ) be a TS.

(a) The set of all covers of (X, ℑ) is denoted by C(X, ℑ) and defined by

C(X,)={BP(X):{B:BB}=X}.

(b) The set of all open covers of (X, ℑ) is denoted by OC(X, ℑ) and defined by

OC(X,)={B:{B:BB}=X}.

We recall that if , where (X, ℑ) is a TS, then is called a refinement of ℬ (denoted as ) if for each , there is B ∈ ℬ such that AB. Moreover, let (X, ℑ) be a TS and ; then, is called locally finite in (X, ℑ) if for each xX, there is U ∈ ℑ such that xU and { : UA ≠ ∅︀} is finite.

Definition 2.2 ([36])

ATS (X, ℑ) is called ω*-paracompact if for every ℬ ∈ OC (X, ℑω), there is ) such that is locally finite in (X, ℑ), and is a refinement of ℬ.

Definition 2.3

Let (X, τ, A) be an STS.

(a) The set of all soft covers of (X, τ, A) is denoted by SC (X, τ, A) and defined by

SC(X,τ,A)={SS(X,A):˜{H:H}=1A}.

(b) The set of all soft open covers of (X, τ, A) is denoted by SOC (X, τ, A) and defined by

SOC(X,τ,A)={τ:˜{H:H}=1A}.

(c) The set of all soft closed covers of (X, τ, A) is denoted by SCC(X, ℑ) and defined by

SCC(X,τ,A)={:τc:˜{H:H}=1A}.

Definition 2.4 ([37])

Let (X, τ, A) be an STS and ℋ, ℳ ⊆ SS(X, A).

(a) ℋ is called soft locally finite in (X, τ, A) if for each axSP(X, A), there is Fτ such that ax ∊̃ F, and {H ∈ ℋ: F ∩̃H ≠ 0A} is finite.

(b) ℳ is called a soft refinement of ℋ (denoted as ℳ ⪯̃ ℋ) if for each M ∈ ℳ, there is H ∈ ℋ such that M ⊂̃ H.

Definition 2.5 ([37])

An STS (X, τ, A) is called soft paracompact if for every ℋ ∈ SOC(X, τ, A), there is ℳ ∈ SOC (X, τ, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.

Definition 2.6 ([4])

An STS (X, τ, A) is called soft regular if for every Fτc and ax∉̃F, there exist U, Vτ such that ax ∊̃ U, F ⊆̃ V, and U ∩̃ V = 0A.

Proposition 2.7 ([4])

An STS (X, τ, A) is soft regular if for every Vτ and ax ∊̃ U, there exists Vτ such that ax ∊̃ V ⊆̃ Clτ (V) ⊆̃ U.

Proposition 2.8 ([37])

Let (X, τ, A) be an STS, where A = {a}. Then, (X, τ, A) is soft paracompact if and only if the TS (X, τa) is paracompact.

Proposition 2.9 ([2])

Let (X, τ, A) be an STS. Then, for all aA, we have (τa)ω = (τω)a.

Proposition 2.10 ([2])

Let X be an initial universe, let A be a set of parameters, and let {ℑa : aA} be an indexed family of topologies on X. Then, (aAa)ω=aA(a)ω.

3. Soft ω*-Paracompactness

This study is primarily concerned with the following concept.

Definition 3.1

An STS (X, τ, A) is called soft ω*-paracompact if for every ℋ ∈ SOC (X, τω, A), there is ℳ ∈ SOC (X, τω, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.

The following three Lemmas are used in Theorem 3.6 below.

Lemma 3.2

If ℋ is soft locally finite in (X, τ, A), then {Clτω (H) : H ∈ ℋ} is soft locally finite in (X, τ, A).

Proof

Let axSP(X, A); then, there is Fτ such that axF, and {H ∈ ℋ: F∩̃H ≠ 0A} is finite.

Claim

{H ∈ ℋ: F∩̃Clτω (H) ≠ 0A} ⊆ {H ∈ ℋ: F∩̃H ≠ 0A}.

Proof of Claim

We assume that F ∩̃ Clτω (H) ≠ 0A for some H ∈ ℋ. We select ax ∊̃ F ∩̃ Clτω (H) ⊂̃ F ∩̃ Clτ (H); then, ax ∊̃ F ∩̃ Clτ (H), and thus F ∩̃ H ≠ 0A.

Lemma 3.3

If ℋ is soft locally finite in (X, τ, A), then ℋ is soft locally finite in (X, τω, A).

Proof

Let axSP(X, A); then, there is Fττω such that axF, and {H ∈ ℋ: F∩̃H ≠ 0A} is finite.

The following example shows that the converse of the implication in Lemma 3.3 is not true in general.

Example 3.4

Let X = ℕ, τ = {0A, 1A}, A = ℝ, and ℋ = SP(X, A). Then, ℋ is soft locally finite in (X, τω, A), but it is not soft locally finite in (X, τ, A).

Lemma 3.5

If ℋ is soft locally finite in (X, τ, A), then

˜{Clτω(H):H}=Clτω(˜{H:H}).
Proof

We assume that ℋ is soft locally finite in (X, τ, A). By Lemma 3.3, ℋ is soft locally finite in (X, τω, A). Thus, by applying Proposition 5.2 in [37], we obtain the result.

The following provides two characterizations of soft ω*-paracompact STSs.

Theorem 3.6

Let (X, τ, A) be an STS so that (X, τω, A) is soft regular. Then, the following are equivalent:

(a) (X, τ, A) is soft ω*-paracompact.

(b) For every ℋ ∈ SOC(X, τω, A), there is ℳ ∈ SC(X, τω, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.

(c) For every ℋ ∈ SOC(X, τω, A), there is ℳ ∈ SCC(X, τω, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.

Proof

(a) ⇒ (b): Obvious.

(b) ⇒ (c): We assume that (X, τ, A) is soft ω*-paracompact, and let ℋ ∈ SOC (X, τω, A). For every axSP(X, A), we select Hax ∈ ℋ such that ax ∊̃ Hax. As (X, τω, A) is soft regular, by Proposition 2.7, for every axSP(X, A), there is Naxτ such that ax ∊̃ Nax ⊂̃ Clτω (Nax) ⊂̃ Hax. Let ; then, . As (X, τ, A) is soft ω*-paracompact, there is ℳ ∈ SOC(X, τω, A) such that ℳ is soft locally finite in (X, τ, A) and . Letℳ1 = {Clτω (M) : M ∈ ℳ}. Then, ℳ1SCC(X, τω, A), and by Lemma 3.2, ℳ1 is soft locally finite.

Claim (i)

1 ⪯̃ ℋ.

Proof of Claim

Let Clτω (M) ∈ ℳ1, where M ∈ ℳ. As , there is axSP(X, A) such that M ⊂̃ Nax, and thus Clτω (M) ⊂̃ Clτω (Nax) ⊂̃ Hax with Hax ∈ ℋ. It follows thatℳ1 ⪯̃ ℋ.

(c) ⇒ (a): Let ℋ ∈ SOC (X, τω, A). By (c), there is ℒ ∈ SCC (X, τω, A) such that ℒ is soft locally finite in (X, τ, A) and ℒ ⪯̃ ℋ. As ℒ is soft locally finite, for each axSP(X, A), there is Naxτ such that ax ∊̃ Nax, and {L ∈ ℒ: Nax ∩̃L ≠ 0A} is finite. Let . Then, , and by (c), there is such that is soft locally finite in (X, τ, A) and . As ℒ ⪯̃ ℋ, for every L ∈ ℒ, there is HL ∈ ℋ such that L ⊂̃ HL. For every L ∈ ℒ, let and ML = KL ∩̃ HL.

Claim (ii)

(1) {KL : L ∈ ℒ}. ⊆ τω.

(2) For every L ∈ ℒ, we have L ⊆̃ KL.

(3) For every L ∈ ℒ and , KL ∩̃ G ≠ 0A if and only if L ∩̃ G ≠ 0A.

(4) {KL : L ∈ ℒ} is soft locally finite.

(5) {ML : L ∈ ℒ} ∈ SOC (X, τω, A).

(6) {ML : L ∈ ℒ} ⪯̃ ℋ.

(7) {ML : L ∈ ℒ} is soft locally finite.

Proof of Claim

1. Follows from Lemma 3.5.

2. We assume toward a contradiction that there are L0 ∈ ℒ and axSP(X, τ) such that . Then, ax ∊̃ L0 and . As , there is such that Gx ∩̃ L0 = 0A and ax ∊̃ Gx. However, ax ∊̃ Gx ∩̃ L0, which is a contradiction.

3. Necessity. Let L ∈ ℒ and such that KL ∩̃ G ≠ 0A, and we assume toward a contradiction that L ∩̃ G = 0A. We select ax ∊̃ KL ∩̃ G. Then, ax ∊̃ KL and ax ∊̃ G. As L ∩̃ G = 0A and ax ∊̃ G, we have and so ax ∉̃ KL, which is a contradiction.

Sufficiency

Let L ∈ ℒ and such that L ∩̃ G ≠ 0A. By (2), L ⊂̃ KL, and therefore L ∩̃ G ⊂̃ KL ∩̃ G. It follows that KL ∩̃ G ≠ 0A.

4. Let axSP(X, τ). As is soft locally finite, there is Sτ such that ax ∊̃ S, and { : G∩̃S ≠ 0A} is finite; let { : G∩̃S ≠ 0A} = {G1, G2, …, Gn}. We assume that for some L ∈ ℒ, we have KL ∩̃ S ≠ 0A. We select by ∊̃ KL ∩̃ S. As by ∊̃ 1A = ∪̃ {G : }, there is such that by ∊̃ G. As by ∊̃ KL, by (3), we have G ∩̃ L ≠ 0A. As by ∊̃ G ∩̃ S, we have G = Gi for some i = 1, 2, …, n. As , for every i = 1, 2, …, n, there is (ai)xi such that Gi ⊂̃ N(ai)xi. Therefore, {L ∈ ℒ : KL∩̃S ≠ 0A} is finite. It follows that {KL : L ∈ ℒ} is soft locally finite.

5. Let L ∈ ℒ. As HL ∈ ℋ ⊆ τω, and by (1), we have KLτω, it follows that ML = KL ∩̃ HLτω. However, by (2) L. Moreover, L ⊂̃ HL, and by (2), we have L ⊂̃ KL; accordingly, L ⊂̃ ML. It follows that {ML : L ∈ ℒ} ⊆ τω and 1A = ∪̃ {L : L ∈ ℒ} ⊂̃ ∪̃ {ML : L ∈ ℒ}. Therefore, {ML : L ∈ ℒ} ∈ SOC(X, τω, A).

6. Let L ∈ ℒ. Then, ML = KL ∩̃ HL ⊂̃ HL with HL ∈ ℋ. Thus, {ML : L ∈ ℒ} ⪯̃ ℋ.

7. This follows directly from (4).

The following characterization of soft paracompact STSs is used in the proof of Theorem 3.8.

Proposition 3.7

Let (X, τ, A) be an STS, and let ℬ be a soft base of τ. Then, (X, τ, A) is soft paracompact if and only if for every ℋ ∈ SOC(X, τ, A) with ℋ ⊆ ℬ, there is ℳ ∈ SOC (X, τ, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.

Proof. Straightforward.

Theorem 3.8

Let X be an initial universe, and let A be a set of parameters. Moreover, let {ℑa : aA} be an indexed family of topologies on X. Then, (X,aAa,A) is soft paracompact if and only if (X, ℑa) is paracompact for all aA.

Proof. Necessity

We assume that (X,aAa,A) is soft paracompact. Let bA. To show that (X, ℑb) is paracompact, let . We set . Then, SOC(X,aAa,A). As (X,aAa,A) is soft paracompact, there is SOC(X,aAa,A) such that ℳ is soft locally finite in (X,aAa,A) and ℳ ⪯̃ ℋ. Let ℳb = {M(b) : M ∈ ℳ}.

Claim

(1) ℳbOC(X, ℑb).

(2) ℳb is locally finite in (X, ℑb).

(3) .

Proof of Claim

1. As aAa, by the definition of ℑb, we have ℳb ⊆ ℑb. As SOC(X,aAa,A), we have ∪̃{M : M ∈ ℳ} = 1A, and thus (∪̃{M : M ∈ ℳ}) (b) = ∪{M(b) : M ∈ ℳ} = X. Therefore, ℳbOC(X, ℑb).

2. Let xX. Then, bxSP(X, A). As ℳ is soft locally finite in (X,aAa,A), there is FaAasuch that bx ∊̃ F, and {M ∈ ℳ: F∩̃M ≠ 0A} is finite. Thus, xF(b) ∈ ℑb. If M ∈ ℳ is such that F(b) ∩ M(b) ≠ ∅︀, then F ∩̃ M ≠ 0A. It follows that ℳb is locally finite in (X, ℑb).

3. Let M(b) ∈ ℳb − {∅︀}, where M ∈ ℳ. As ℳ ⪯̃ ℋ, there is H ∈ ℋ such that M ⊂̃ H, and thus M(b) ⊆ H(b). As M(b) ≠ ∅︀, we have H(b) ≠ ∅︀, and thus there is such that H = bU and H(b) = U. It follows that .

Sufficiency

We assume that (X, ℑa) is paracompact for all aA. Let ℬ = {aY : aA and Y ∈ ℑa}. By Theorem 3.5 in [1], ℬ is a soft base of aAa. We apply Proposition 3.7. Let SOC(X,aAa,A) with ℋ ⊆ ℬ. For each aA, let ℋa = {YX : aY ∈ ℋ}. Then, for all aA, ℋaOC (X, ℑa), where (X, ℑa) is paracompact, and thus there is ℳaOC (X, ℑa) such that ℳa is locally finite in (X, ℑa) and ℳa ⪯ ℋa. Let .

Claim

(1) GSOC(X,aAa,A).

(2) is soft locally finite in (X,aAa,A).

(3) .

Proof of Claim

1. For all aA, we have ℳa ⊆ ℑa, and therefore GaAa. For all aA, we have ℳaOC (X, ℑa), and thus (∪̃{aY : aA and Y ∈ ℳa}) (a) = ∪{Y : Y ∈ ℳa} = X. Therefore, ∪̃{aY : aA and Y ∈ ℳa} = 1A. It follows that GSOC(X,aAa,A).

2. Let bxSP(X, A). As ℳa is locally finite in (X, ℑb), there is O ∈ ℑb such that xO, and {Y : OY ≠ ∅︀} is finite.

We have bxbOaAa. If bO ∩̃ aY ≠ 0A, then a = b and OY ≠ ∅︀. This shows that {aY : aA and Y ∈ ℳa, and bO∩̃aY ≠ 0A} is finite. It follows that is soft locally finite in (X,aAa,A).

3. Let with aA and Y ∈ ℳa. As ℳa ⪯ ℋa, there is Z ∈ ℋa, where aZ ∈ ℋ, such that YZ. Therefore, aY ⊂̃ aZ. It follows that .

Proposition 3.9

Let (X, τ, A) be an STS, and let ℬ be a soft base of τω. Then, (X, τ, A) is soft ω*-paracompact if and only if for every ℋ ∈ SOC(X, τω, A) with ℋ ⊆ ℬ, there is ℳ ∈ SOC(X, τω, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.

Proof. Straightforward.

Corollary 3.10

Let (X, τ, A) be an STS. Then, (X, τ, A) is soft ω*-paracompact if and only if for every ℋ ∈ SOC(X, τω, A) with ℋ ⊆ {FH : Fτ and HCSS(X, A)}, there is ℳ ∈ SOC (X, τω, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.

Lemma 3.11

Let (X, τ, A) be an STS. If ℬ is a soft base for τ, then {BH : B ∈ ℬ and HCSS(X, A)} is a soft base for τω.

Proof. Straightforward.

Theorem 3.12

Let (X, τ, A) be an STS, where A = {a}. Then, (X, τ, A) is soft ω*-paracompact if and only if the TS (X, τa) is ω*-paracompact.

Proof. Necessity

We assume that (X, τ, A) is soft ω*-paracompact. Let . By Proposition 2.9, we have (τa)ω = (τω)a, and thus .

Then, {aU : } ∈ SOC(X, τω, A). As (X, τ, A) is soft ω*-paracompact, there is ℋ ∈ SOC(X, τω, A) such that ℋ is soft locally finite in (X, τ, A) and . Let .

Claim

(1) .

(2) is locally finite in (X, τa).

(3) .

Proof of Claim

1. As ℋ ⊆ τω, we have and by Proposition 2.9, it follows that . In addition, as ℋ ∈ SOC(X, τω, A), we have ∪̃{H : H ∈ ℋ} = 1A, and thus (∪̃{H : H ∈ ℋ}) (a) = ∪{H (a) : H ∈ ℋ} = X. It follows that .

2. Let xX. Then, axSP(X, A). As ℋ is soft locally finite in (X, τ, A), there is Fτ such that ax ∊̃ F, and {H ∈ ℋ: F∩̃H ≠ 0A} is finite. Then, xF (a) ∈ τa. Furthermore, if for H ∈ ℋ, we have F (a) ∩ H (a) ≠ ∅︀, then F ∩̃ H ≠ 0A. It follows that {H ∈ ℋ: F (a) ∩ H (a) ≠ ∅︀} is finite. Therefore, is locally finite in (X, τa).

3. Let , where H ∈ ℋ. As , there is such that H ⊂̃ aU; thus, H (a) ⊆ U. This shows that .

Sufficiency

We assume that (X, τa) is ω*-paracompact. Let ℋ ∈ SOC(X, τω, A). Then, {H (a) : H ∈ ℋ} ∈ OC(X, (τω)a) = OC(X, (τa)ω). As (X, τa) is ω*-paracompact, there is such that is locally finite in (X, τa) and . Let .

Claim

(1) ℳ ∈ SOC(X, τω, A).

(2) ℳ is soft locally finite in (X, τ, A).

(3) ℳ ⪯̃ ℋ.

Proof of Claim

(1) As , we have ℳ ⊆ τω. In addition, as , we have , and thus . It follows that ℳ ∈ SOC(X, τω, A).

(2) Let axSP(X, A). As is locally finite in (X, τa), there is Oτa such that xO and { : OV ≠ ∅︀} is finite. Then, ax ∊̃ aOτ. If aV ∈ ℳ, where and aO ∩̃ aV ≠ 0A, then (aO∩̃aV) (a) = OV ≠ ∅︀. It follows that ℳ is soft locally finite in SOC(X, τ, A).

(3) Let aV ∈ ℳ, where . As , there is H ∈ ℋ such that VH (a), and thus aV ⊂̃ H. This shows that ℳ ⪯̃ ℋ.

Theorem 3.13

Let X be an initial universe, and let A be a set of parameters. Moreover, let {ℑa : aA} be an indexed family of topologies on X. Then, (X,aAa,A) is soft ω*-paracompact if and only if (X, ℑa) is ω*-paracompact for all aA.

Proof. Necessity

We assume that (X,aAa,A) is soft ω*-paracompact. Let bA. To show that (X, ℑb) is ω*-paracompact, let . We set . Then, SOC(X,aA(a)ω,A). By Proposition 2.10, aA(a)ω=(aAa)ω, and thus SOC(X,(aAa)ω,A). As (X,aAa,A) is soft ω*-paracompact, there is SOC(X,(aAa)ω,A) such that ℳ is soft locally finite in (X,aAa,A) and ℳ ⪯̃ ℋ. By Proposition 2.10, SOC(X,aA(a)ω,A). Let ℳb = {M (b) : M ∈ ℳ}.

Claim

(1) ℳbOC(X, (ℑb)ω).

(2) ℳb is locally finite in (X, ℑb).

(3) .

Proof of Claim

1. As aA(a)ω, by the definition of ℑb, we have ℳb ⊆ (ℑb)ω. As SOC(X,aA(a)ω,A), we have ∪̃{M : M ∈ ℳ} = 1A; hence, ∪̃{M : M ∈ ℳ})(b) = ∪{M(b) : M ∈ ℳ} = X. Therefore, ℳbOC(X, (ℑb)ω).

2. Let xX. Then, bxSP(X, A). As ℳ is soft locally finite in (X,aAa,A), there is FaAa such that bx ∊̃ F, and {M ∈ ℳ: F∩̃M ≠ 0A} is finite. Thus, xF(b) ∈ ℑb. If M ∈ ℳ is such that F(b) ∩ M(b) ≠ ∅︀, then F ∩̃ M ≠ 0A. It follows that ℳb is locally finite in (X, ℑb).

3. Let M(b) ∈ ℳb − {∅︀}, where M ∈ ℳ. As ℳ ⪯̃ ℋ, there is H ∈ ℋ such that M ⊂̃ H, and thus M(b) ⊆ H(b). As M(b) ≠ ∅︀, we have H(b) ≠ ∅︀, and thus there is such that H = bU and H(b) = U. It follows that .

Sufficiency

We assume that (X, ℑa) is ω*-paracompact for all aA. Let ℬ = {aY : aA and Y ∈ (ℑa)ω}. Then, ℬ is a soft base of aA(a)ω=(aAa)ω. We apply Proposition 3.9. Let SOC(X,(aAa)ω,A) with ℋ ⊆ ℬ. For each aA, let ℋa = {YX : aY ∈ ℋ}. Then, for all aA, we have ℋaOC (X, (ℑa)ω,), where (X, ℑa) is ω*-paracompact. Thus, there is ℳaOC (X, (ℑa)ω) such that ℳa is locally finite in (X, ℑa) and ℳa ⪯ ℋa. Let .

Claim

(1) GSOC(X,(aAa)ω,A).

(2) is soft locally finite in (X,aAa,A).

(3) .

Proof of Claim

1. For all aA, we have ℳa ⊆ (ℑa)ω, and therefore GaA(a)ω=(aAa)ω. For all aA, we have ℳaOC (X, (ℑa)ω), and thus (∪̃{aY : aA and Y ∈ ℳa}) (a) = ∪{Y : Y ∈ ℳa} = X. Therefore, ∪̃{aY : aA and Y ∈ ℳa} = 1A. It follows that GSOC(X,(aAa)ω,A).

2. Let bxSP(X, A). As ℳa is locally finite in (X, ℑb), there is O ∈ ℑb such that xO, and {Y : OY ≠ ∅︀} is finite. We have bxbOaAa. If bO ∩̃ aY ≠ 0A, then a = b and OY ≠ ∅︀. This shows that {aY : aA and Y ∈ ℳa, and bO∩̃aY ≠ 0A} is finite. It follows that is soft locally finite in (X,aAa,A).

3. Let with aA and Y ∈ ℳa. As ℳa ⪯ ℋa, there is Z ∈ ℋa, where aZ ∈ ℋ, such that YZ. Therefore, aY ⊂̃ aZ. It follows that .

4. Soft ω*-Paracompactness versus Soft Paracompactness

Herein, we show by examples that soft ω*-paracompactness and soft paracompactness are independent of each other.

Theorem 4.1

Every soft regular soft ω*-paracompact STS is soft paracompact.

Proof

Let (X, τ, A) be soft regular and soft ω*-paracompact. Let ℋ ∈ SOC (X, τ, A) ⊆ SOC (X, τω, A). Then, ℋ ∈ SOC(X, τω, A). As (X, τ, A) is soft ω*-paracompact, there is ℳ ∈ SOC(X, τω, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ. It is clear that ℳ ∈ SC (X, τ, A). As (X, τ, A) is soft regular, by Theorem 5.14 in [37], it follows that (X, τ, A) is soft paracompact.

Theorem 4.2

If (X, τ, A) is a soft ω*-paracompact STS, then (X, τω, A) is soft paracompact.

Proof

We assume that (X, τ, A) is soft ω*-paracompact. Let ℋ ∈ SOC(X, τω, A). As (X, τ, A) is soft ω*-paracompact, there is ℳ ∈ SOC(X, τω, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ. As ℳ is soft locally finite in (X, τ, A), we have that ℳ is soft locally finite in (X, τω, A). It follows that (X, τω, A) is soft paracompact.

Definition 4.3

An STS (X, τ, A) is called strongly soft locally finite if for every bxSP(X, A), there exists Hτ such that Supp (H) is finite, bx∊̃H, and H (a) is finite for all aA.

Lemma 4.4

Let (X, τ, A) be a strongly soft locally finite STS, and let ℋ ⊆SP(X, A). Then, ℋ is soft locally finite in (X, τ, A).

Proof

We assume that (X, τ, A) is a strongly soft locally finite, and let ℋ ⊆SP(X, A). Let bxSP(X, A). As (X, τ, A) is strongly soft locally finite, there exists Gτ such that Supp (G) is finite, bx∊̃G, and G(a) is finite for all aA. It is not difficult to verify that {ax : ax∊̃G} is finite. Thus, {H ∈ ℋ: G∩̃H ≠ 0A} is finite. It follows that ℋ is soft locally finite in (X, τ, A).

Theorem 4.5

Every strongly soft locally finite STS is soft ω*-paracompact.

Proof

We assume that (X, τ, A) is strongly soft locally finite. Let ℋ ∈SOC(X, τω, A). Let ℳ= SP(X, A). As (X, τ, A) is strongly soft locally finite, (X, τ, A) is soft locally countable, and by Corollary 14 in [2], we have that (X, τω, A) is a discrete STS. Thus, ℳ ∈ SOC(X, τω, A). By Lemma 4.4, ℳ is soft locally finite in (X, τ, A). Moreover, it is clear that ℳ ⪯̃ ℋ. It follows that (X, τω, A) is soft ω*-paracompact.

Theorem 4.6

Every strongly soft locally countable soft ω*-paracompact STS is strongly soft locally finite.

Proof

Let (X, τ, A) be strongly soft locally countable and soft ω*-paracompact. Let ℋ = SP(X, A). Then, by Corollary 14 in [2], (X, τω, A) is a discrete STS, and thus ℋ ∈ SOC(X, τω, A). As (X, τ, A) is soft ω*-paracompact, there is ℳ ∈ SOC(X, τω, A) such that ℳ is soft locally finite in (X, τ, A),ℳ ⪯̃ ℋ, and for every M ∈ ℳ, we have M ≠ 0A. For every M ∈ ℳ, there is axM such that M ⊂̃ axM, and thus M = axM. Therefore, ℳ ⊆ SP(X, A). As ℳ ∈ SOC(X, τω, A), we have ℳ = SP(X, A). As ℳ is soft locally finite in (X, τ, A), for every axSP(X, A), there is Gτ such that ax ∊̃ G, and {M ∈ ℳ: G∩̃M ≠ 0A} = {bySP(X, A): G∩̃by ≠ 0A} is finite; thus, Supp (G) is finite, and for every bA, G(b) is finite. It follows that (X, τ, A) is strongly soft locally finite.

Theorem 4.7

If (X, τ, A) is soft separable and soft anti-locally countable, then (X, τω, A) is not soft regular.

Proof

We assume toward a contradiction that (X, τω, A) is soft regular. As (X, τ, A) is soft separable, there is GSCSS(X, A) such that 1A = Clτ (G). As (X, τ, A) is strongly soft anti-locally countable, we have 1ASCSS (X, A); thus, ax ∊̃ 1AG. As (X, τω, A) is soft regular and G is soft closed in (X, τω, A), there exist Uτω, Vτ and HCSS(X, A) such that G ⊂̃ U, ax ∊̃ VH, and U ∩̃ (VH) = 0A. As 1A = Clτ (G) and ax ∊̃ Vτ, there is by∊̃ G ∩̃ V. As by ∊̃ G ⊂̃ U, there exist Wτ and MCSS(X, A) such that by ∊̃ WM ⊂̃ U. As by ∊̃ U∩̃ Wτ, we have U∩̃ Wτ − {0A}. As (X, τ, A) is soft anti-locally countable, we have U∩̃ WCSS(X, A). As U∩̃ WH∩̃G ⊂̃ U ∩̃ (VH) = 0A, we have U∩̃ W ⊂̃ H∩̃G, and thus U∩̃ WCSS (X, A), which is a contradiction.

Theorem 4.8

Every soft anti-locally countable soft T2 soft paracompact soft separable STS is not soft ω*-paracompact.

Proof

We assume that (X, τ, A) is soft anti-locally countable, soft T2, soft ω*-paracompact, and soft separable. We assume toward a contradiction that (X, τ, A) is soft ω*-paracompact. As (X, τ, A) is soft ω*-paracompact, by Theorem 4.2 (X, τω, A) is soft paracompact. As (X, τ, A) is soft T2, it is clear that (X, τω, A) is soft T2. As (X, τω, A) is soft T2 and soft paracompact, by Theorem 5.8 in [37], (X, τω, A) is soft regular. However, as (X, τ, A) is soft anti-locally countable and soft separable, by Theorem 4.1, (X, τω, A) is not soft regular, which is a contradiction.

Example 4.9

Let A = {a}, X = ℝ, and τ = {FSS(X, A) : F(a) belongs to the usual topology on ℝ}. Then, clearly, (X, τ, A) is soft anti-locally countable, soft T2, soft paracompact, and soft separable. Thus, by Theorem 4.8, (X, τ, A) is not soft ω*-paracompact.

Example 4.10

Let (X, ℑ) be the irrational slope topology (Example 75 in [38]). Let A{a} and τ = {FSS(X, A) : F(a) ∈ ℑ}. Clearly, (X, τω, A) is a discrete STS. However, by Theorem 4.6, (X, τω, A) is not soft ω*-paracompact.

5. Conclusion

We defined and studied a new concept STSs, soft ω*-para- compactness. Several characterizations and relationships regarding this new concept were proved. In the future, we intend to provide more characterizations and prove soft mapping theorems for soft ω*-paracompactness, as well as define new soft topological concepts using soft ω-open sets.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.


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Biography

Samer Al Ghour received the Ph.D. in Mathematics from University of Jordan, Jordan in 1999. Currently, he is a professor at the Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan. His research interests include general topology, fuzzy topology, and soft set theory.

E-mail: algore@just.edu.jo