Soft ω^{*}-Paracompactness in Soft Topological Spaces

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Received October 10, 2020; Revised January 6, 2021; Accepted January 19, 2021.

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- 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*G*∈*SS*(*X*,*A*) is such that*G*(*a*) is a countable set for all*a*∈*X*, 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 0and 1_{A}, respectively. Soft topological spaces were defined in [4] as follows: A triplet (_{A}*X*,*τ*,*A*), where*τ*⊆*SS*(*X*,*A*), is called an STS if 0and 1_{A}∈_{A}*τ*, 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, 5–28]. 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*U*⊆*X*; then,*U*is called*ω*-open if for every*x*∈*U*, there is an open set*V*and a countable subset*C*⊆*X*such that*x*∈*V*−*C*⊆*U*.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 ℑ. It is known that ℑ_{ω}forms a topology on_{ω}*X*that contains ℑ. The study of*ω*-open sets remains an area that attracts considerable attention [30–35]. 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*G*∈*SS*(*X*,*A*); then,*G*is called a soft*ω*-open set if for all*a*∊̃_{x}*H*, there exist*F*∈*τ*and*H*∈*CSS*(*X*,*A*) such that*a*∊̃_{x}*F*−*H*⊂̃*G*. The collection of all soft*ω*-open sets in (*X*,*τ*,*A*) is denoted by*τ*. It was proved in [2] that (_{ω}*X*,*τ*,_{ω}*A*) is an STS with*τ*⊆*τ*._{ω}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$$C(X,\Im )=\{\mathcal{B}\subseteq \mathcal{P}(X):\cup \{B:B\in \mathcal{B}\}=X\}.$$

(b) The set of all open covers of (

*X*, ℑ) is denoted by*OC*(*X*, ℑ) and defined by$$OC(X,\Im )=\{\mathcal{B}\subseteq \Im :\cup \{B:B\in \mathcal{B}\}=X\}.$$

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*A*⊆*B*. Moreover, let (*X*, ℑ) be a TS and ; then, is called locally finite in (*X*, ℑ) if for each*x*∈*X*, there is*U*∈ ℑ such that*x*∈*U*and { :*U*∩*A*≠ ∅︀} is finite.### Definition 2.2 ([36])

ATS (

*X*, ℑ) is called*ω*^{*}-paracompact if for every ℬ ∈*OC*(*X*, ℑ), there is ) such that is locally finite in (_{ω}*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$$SC\hspace{0.17em}(X,\tau ,A)=\{\mathscr{H}\subseteq SS\hspace{0.17em}(X,A):\tilde{\cup}\{H:H\in \mathscr{H}\}={1}_{A}\}.$$

(b) The set of all soft open covers of (

*X*,*τ*,*A*) is denoted by*SOC*(*X*,*τ*,*A*) and defined by$$SOC\hspace{0.17em}(X,\tau ,A)=\{\mathscr{H}\subseteq \tau :\tilde{\cup}\{H:H\in \mathscr{H}\}={1}_{A}\}.$$

(c) The set of all soft closed covers of (

*X*,*τ*,*A*) is denoted by*SCC*(*X*, ℑ) and defined by$$SCC\hspace{0.17em}(X,\tau ,A)=\{\mathscr{H}:\mathscr{H}\subseteq {\tau}^{c}:\tilde{\cup}\{H:H\in \mathscr{H}\}={1}_{A}\}.$$

### 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*a*∈_{x}*SP*(*X*,*A*), there is*F*∈*τ*such that*a*∊̃_{x}*F*, and {*H*∈ ℋ:*F*∩̃*H*≠ 0} is finite._{A}(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*F*∈*τ*and^{c}*a*∉̃_{x}*F*, there exist*U*,*V*∈*τ*such that*a*∊̃_{x}*U*,*F*⊆̃*V*, and*U*∩̃*V*= 0._{A}### Proposition 2.7 ([4])

An STS (

*X*,*τ*,*A*) is soft regular if for every*V*∈*τ*and*a*∊̃_{x}*U*, there exists*V*∈*τ*such that*a*∊̃_{x}*V*⊆̃*Cl*(_{τ}*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*,*τ*) is paracompact._{a}### Proposition 2.9 ([2])

Let (

*X*,*τ*,*A*) be an STS. Then, for all*a*∈*A*, we have (*τ*)_{a}= (_{ω}*τ*)_{ω}._{a}### Proposition 2.10 ([2])

Let

*X*be an initial universe, let*A*be a set of parameters, and let {ℑ:_{a}*a*∈*A*} be an indexed family of topologies on*X*. Then,${\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega}={\displaystyle \underset{a\in A}{\oplus}}{\left({\Im}_{a}\right)}_{\omega}$ .

- 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*,*τ*,_{ω}*A*), there is ℳ ∈*SOC*(*X*,*τ*,_{ω}*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 {*Cl*_{τ}_{ω}(*H*) :*H*∈ ℋ} is soft locally finite in (*X*,*τ*,*A*).**Proof**Let

*a*∈_{x}*SP*(*X*,*A*); then, there is*F*∈*τ*such that*a*∈_{x}*F*, and {*H*∈ ℋ:*F*∩̃*H*≠ 0} is finite._{A}### Claim

{

*H*∈ ℋ:*F*∩̃*Cl*_{τ}_{ω}(*H*) ≠ 0} ⊆ {_{A}*H*∈ ℋ:*F*∩̃*H*≠ 0}._{A}### Proof of Claim

We assume that

*F*∩̃*Cl*_{τ}_{ω}(*H*) ≠ 0for some_{A}*H*∈ ℋ. We select*a*∊̃_{x}*F*∩̃*Cl*_{τ}_{ω}(*H*) ⊂̃*F*∩̃*Cl*(_{τ}*H*); then,*a*∊̃_{x}*F*∩̃*Cl*(_{τ}*H*), and thus*F*∩̃*H*≠ 0._{A}### Lemma 3.3

If ℋ is soft locally finite in (

*X*,*τ*,*A*), then ℋ is soft locally finite in (*X*,*τ*,_{ω}*A*).**Proof**Let

*a*∈_{x}*SP*(*X*,*A*); then, there is*F*∈*τ*⊆*τ*such that_{ω}*a*∈_{x}*F*, and {*H*∈ ℋ:*F*∩̃*H*≠ 0} is finite._{A}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*= {0, 1_{A}},_{A}*A*= ℝ, and ℋ =*SP*(*X*,*A*). Then, ℋ is soft locally finite in (*X*,*τ*,_{ω}*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$$\tilde{\cup}\{{Cl}_{{\tau}_{\omega}}\hspace{0.17em}(H):H\in \mathscr{H}\}={Cl}_{{\tau}_{\omega}}\hspace{0.17em}(\tilde{\cup}\{H:H\in \mathscr{H}\}).$$

**Proof**We assume that ℋ is soft locally finite in (

*X*,*τ*,*A*). By Lemma 3.3, ℋ is soft locally finite in (*X*,*τ*,_{ω}*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*,*τ*,_{ω}*A*) is soft regular. Then, the following are equivalent:(a) (

*X*,*τ*,*A*) is soft*ω*^{*}-paracompact.(b) For every ℋ ∈

*SOC*(*X*,*τ*,_{ω}*A*), there is ℳ ∈*SC*(*X*,*τ*,_{ω}*A*) such that ℳ is soft locally finite in (*X*,*τ*,*A*) and ℳ ⪯̃ ℋ.(c) For every ℋ ∈

*SOC*(*X*,*τ*,_{ω}*A*), there is ℳ ∈*SCC*(*X*,*τ*,_{ω}*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*,*τ*,_{ω}*A*). For every*a*∈_{x}*SP*(*X*,*A*), we select*H*_{a}_{x}∈ ℋ such that*a*∊̃_{x}*H*_{a}_{x}. As (*X*,*τ*,_{ω}*A*) is soft regular, by Proposition 2.7, for every*a*∈_{x}*SP*(*X*,*A*), there is*N*_{a}_{x}∈*τ*such that*a*∊̃_{x}*N*_{a}_{x}⊂̃*Cl*_{τ}_{ω}(*N*_{a}_{x}) ⊂̃*H*_{a}_{x}. Let ; then, . As (*X*,*τ*,*A*) is soft*ω*^{*}-paracompact, there is ℳ ∈*SOC*(*X*,*τ*,_{ω}*A*) such that ℳ is soft locally finite in (*X*,*τ*,*A*) and . Letℳ_{1}= {*Cl*_{τ}_{ω}(*M*) :*M*∈ ℳ}. Then, ℳ_{1}∈*SCC*(*X*,*τ*,_{ω}*A*), and by Lemma 3.2, ℳ_{1}is soft locally finite.### Claim (i)

ℳ

_{1}⪯̃ ℋ.### Proof of Claim

Let

*Cl*_{τ}_{ω}(*M*) ∈ ℳ_{1}, where*M*∈ ℳ. As , there is*a*∈_{x}*SP*(*X*,*A*) such that*M*⊂̃*N*_{a}_{x}, and thus*Cl*_{τ}_{ω}(*M*) ⊂̃*Cl*_{τ}_{ω}(*N*_{a}_{x}) ⊂̃*H*_{a}_{x}with*H*_{a}_{x}∈ ℋ. It follows thatℳ_{1}⪯̃ ℋ.(c) ⇒ (a): Let ℋ ∈

*SOC*(*X*,*τ*,_{ω}*A*). By (c), there is ℒ ∈*SCC*(*X*,*τ*,_{ω}*A*) such that ℒ is soft locally finite in (*X*,*τ*,*A*) and ℒ ⪯̃ ℋ. As ℒ is soft locally finite, for each*a*∈_{x}*SP*(*X*,*A*), there is*N*_{a}_{x}∈*τ*such that*a*∊̃_{x}*N*_{a}_{x}, and {*L*∈ ℒ:*N*_{a}_{x}∩̃*L*≠ 0} is finite. Let . Then, , and by (c), there is such that is soft locally finite in (_{A}*X*,*τ*,*A*) and . As ℒ ⪯̃ ℋ, for every*L*∈ ℒ, there is*H*∈ ℋ such that_{L}*L*⊂̃*H*. For every_{L}*L*∈ ℒ, let and*M*=_{L}*K*∩̃_{L}*H*._{L}### Claim (ii)

(1) {

*K*:_{L}*L*∈ ℒ}. ⊆*τ*._{ω}(2) For every

*L*∈ ℒ, we have*L*⊆̃*K*._{L}(3) For every

*L*∈ ℒ and ,*K*∩̃_{L}*G*≠ 0if and only if_{A}*L*∩̃*G*≠ 0._{A}(4) {

*K*:_{L}*L*∈ ℒ} is soft locally finite.(5) {

*M*:_{L}*L*∈ ℒ} ∈*SOC*(*X*,*τ*,_{ω}*A*).(6) {

*M*:_{L}*L*∈ ℒ} ⪯̃ ℋ.(7) {

*M*:_{L}*L*∈ ℒ} is soft locally finite.### Proof of Claim

1. Follows from Lemma 3.5.

2. We assume toward a contradiction that there are

*L*_{0}∈ ℒ and*a*∈_{x}*SP*(*X*,*τ*) such that . Then,*a*∊̃_{x}*L*_{0}and . As , there is such that*G*∩̃_{x}*L*_{0}= 0and_{A}*a*∊̃_{x}*G*. However,_{x}*a*∊̃_{x}*G*∩̃_{x}*L*_{0}, which is a contradiction.3.

*Necessity*. Let*L*∈ ℒ and such that*K*∩̃_{L}*G*≠ 0, and we assume toward a contradiction that_{A}*L*∩̃*G*= 0. We select_{A}*a*∊̃_{x}*K*∩̃_{L}*G*. Then,*a*∊̃_{x}*K*and_{L}*a*∊̃_{x}*G*. As*L*∩̃*G*= 0and_{A}*a*∊̃_{x}*G*, we have and so*a*∉̃_{x}*K*, which is a contradiction._{L}**Sufficiency**Let

*L*∈ ℒ and such that*L*∩̃*G*≠ 0. By (2),_{A}*L*⊂̃*K*, and therefore_{L}*L*∩̃*G*⊂̃*K*∩̃_{L}*G*. It follows that*K*∩̃_{L}*G*≠ 0._{A}4. Let

*a*∈_{x}*SP*(*X*,*τ*). As is soft locally finite, there is*S*∈*τ*such that*a*∊̃_{x}*S*, and { :*G*∩̃*S*≠ 0} is finite; let { :_{A}*G*∩̃*S*≠ 0} = {_{A}*G*_{1},*G*_{2}, …,*G*}. We assume that for some_{n}*L*∈ ℒ, we have*K*∩̃_{L}*S*≠ 0. We select_{A}*b*∊̃_{y}*K*∩̃_{L}*S*. As*b*∊̃ 1_{y}= ∪̃ {_{A}*G*: }, there is such that*b*∊̃_{y}*G*. As*b*∊̃_{y}*K*, by (3), we have_{L}*G*∩̃*L*≠ 0. As_{A}*b*∊̃_{y}*G*∩̃*S*, we have*G*=*G*for some_{i}*i*= 1, 2, …,*n*. As , for every*i*= 1, 2, …,*n*, there is (*a*)_{i}_{x}_{i}such that*G*⊂̃_{i}*N*_{(})_{ai}. Therefore, {_{xi}*L*∈ ℒ :*K*∩̃_{L}*S*≠ 0} is finite. It follows that {_{A}*K*:_{L}*L*∈ ℒ} is soft locally finite.5. Let

*L*∈ ℒ. As*H*∈ ℋ ⊆_{L}*τ*, and by (1), we have_{ω}*K*∈_{L}*τ*, it follows that_{ω}*M*=_{L}*K*∩̃_{L}*H*∈_{L}*τ*. However, by (2)_{ω}*L*. Moreover,*L*⊂̃*H*, and by (2), we have_{L}*L*⊂̃*K*; accordingly,_{L}*L*⊂̃*M*. It follows that {_{L}*M*:_{L}*L*∈ ℒ} ⊆*τ*and 1_{ω}= ∪̃ {_{A}*L*:*L*∈ ℒ} ⊂̃ ∪̃ {*M*:_{L}*L*∈ ℒ}. Therefore, {*M*:_{L}*L*∈ ℒ} ∈*SOC*(*X*,*τ*,_{ω}*A*).6. Let

*L*∈ ℒ. Then,*M*=_{L}*K*∩̃_{L}*H*⊂̃_{L}*H*with_{L}*H*∈ ℋ. Thus, {_{L}*M*:_{L}*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 {ℑ:_{a}*a*∈*A*} be an indexed family of topologies on*X*. Then,$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft paracompact if and only if (*X*, ℑ) is paracompact for all_{a}*a*∈*A*.**Proof. Necessity**We assume that

$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft paracompact. Let*b*∈*A*. To show that (*X*, ℑ) is paracompact, let . We set . Then,_{b}$\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ . As$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft paracompact, there is$\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ such that ℳ is soft locally finite in$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ and ℳ ⪯̃ ℋ. Let ℳ= {_{b}*M*(*b*) :*M*∈ ℳ}.**Claim**(1) ℳ

∈_{b}*OC*(*X*, ℑ)._{b}(2) ℳ

is locally finite in (_{b}*X*, ℑ)._{b}(3) .

### Proof of Claim

1. As

$\mathcal{M}\subseteq {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$ , by the definition of ℑ, we have ℳ_{b}⊆ ℑ_{b}. As_{b}$\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ , we have ∪̃{*M*:*M*∈ ℳ} = 1, and thus (∪̃{_{A}*M*:*M*∈ ℳ}) (*b*) = ∪{*M*(*b*) :*M*∈ ℳ} =*X*. Therefore, ℳ∈_{b}*OC*(*X*, ℑ)._{b}2. Let

*x*∈*X*. Then,*b*∈_{x}*SP*(*X*,*A*). As ℳ is soft locally finite in$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ , there is$F\in {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$ such that*b*∊̃_{x}*F*, and {*M*∈ ℳ:*F*∩̃*M*≠ 0} is finite. Thus,_{A}*x*∈*F*(*b*) ∈ ℑ. If_{b}*M*∈ ℳ is such that*F*(*b*) ∩*M*(*b*) ≠ ∅︀, then*F*∩̃*M*≠ 0. It follows that ℳ_{A}is locally finite in (_{b}*X*, ℑ)._{b}3. Let

*M*(*b*) ∈ ℳ− {∅︀}, where_{b}*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*=*b*and_{U}*H*(*b*) =*U*. It follows that .### Sufficiency

We assume that (

*X*, ℑ) is paracompact for all_{a}*a*∈*A*. Let ℬ = {*a*:_{Y}*a*∈*A*and*Y*∈ ℑ}. By Theorem 3.5 in [1], ℬ is a soft base of_{a}$\underset{a\in A}{\oplus}}{\Im}_{a$ . We apply Proposition 3.7. Let$\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ with ℋ ⊆ ℬ. For each*a*∈*A*, let ℋ= {_{a}*Y*⊆*X*:*a*∈ ℋ}. Then, for all_{Y}*a*∈*A*, ℋ∈_{a}*OC*(*X*, ℑ), where (_{a}*X*, ℑ) is paracompact, and thus there is ℳ_{a}∈_{a}*OC*(*X*, ℑ) such that ℳ_{a}is locally finite in (_{a}*X*, ℑ) and ℳ_{a}⪯ ℋ_{a}. Let ._{a}**Claim**(1)

$\mathcal{G}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ .(2) is soft locally finite in

$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ .(3) .

### Proof of Claim

1. For all

*a*∈*A*, we have ℳ⊆ ℑ_{a}, and therefore_{a}$\mathcal{G}\subseteq {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$ . For all*a*∈*A*, we have ℳ∈_{a}*OC*(*X*, ℑ), and thus (∪̃{_{a}*a*:_{Y}*a*∈*A*and*Y*∈ ℳ}) (_{a}*a*) = ∪{*Y*:*Y*∈ ℳ} =_{a}*X*. Therefore, ∪̃{*a*:_{Y}*a*∈*A*and*Y*∈ ℳ} = 1_{a}. It follows that_{A}$\mathcal{G}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ .2. Let

*b*∈_{x}*SP*(*X*,*A*). As ℳis locally finite in (_{a}*X*, ℑ), there is_{b}*O*∈ ℑsuch that_{b}*x*∈*O*, and {*Y*:*O*∩*Y*≠ ∅︀} is finite.We have

${b}_{x}\in {b}_{O}\in {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$ . If*b*∩̃_{O}*a*≠ 0_{Y}, then_{A}*a*=*b*and*O*∩*Y*≠ ∅︀. This shows that {*a*:_{Y}*a*∈*A*and*Y*∈ ℳ, and_{a}*b*∩̃_{O}*a*≠ 0_{Y}} is finite. It follows that is soft locally finite in_{A}$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ .3. Let with

*a*∈*A*and*Y*∈ ℳ. As ℳ_{a}⪯ ℋ_{a}, there is_{a}*Z*∈ ℋ, where_{a}*a*∈ ℋ, such that_{Z}*Y*⊆*Z*. Therefore,*a*⊂̃_{Y}*a*. It follows that ._{Z}### Proposition 3.9

Let (

*X*,*τ*,*A*) be an STS, and let ℬ be a soft base of*τ*. Then, (_{ω}*X*,*τ*,*A*) is soft*ω*^{*}-paracompact if and only if for every ℋ ∈*SOC*(*X*,*τ*,_{ω}*A*) with ℋ ⊆ ℬ, there is ℳ ∈*SOC*(*X*,*τ*,_{ω}*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*,*τ*,_{ω}*A*) with ℋ ⊆ {*F*−*H*:*F*∈*τ*and*H*∈*CSS*(*X*,*A*)}, there is ℳ ∈*SOC*(*X*,*τ*,_{ω}*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 {*B*−*H*:*B*∈ ℬ and*H*∈*CSS*(*X*,*A*)} is a soft base for*τ*._{ω}**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*,*τ*) is_{a}*ω*^{*}-paracompact.**Proof. Necessity**We assume that (

*X*,*τ*,*A*) is soft*ω*^{*}-paracompact. Let . By Proposition 2.9, we have (*τ*)_{a}= (_{ω}*τ*)_{ω}, and thus ._{a}Then, {

*a*: } ∈_{U}*SOC*(*X*,*τ*,_{ω}*A*). As (*X*,*τ*,*A*) is soft*ω*^{*}-paracompact, there is ℋ ∈*SOC*(*X*,*τ*,_{ω}*A*) such that ℋ is soft locally finite in (*X*,*τ*,*A*) and . Let .**Claim**(1) .

(2) is locally finite in (

*X*,*τ*)._{a}(3) .

### Proof of Claim

1. As ℋ ⊆

*τ*, we have and by Proposition 2.9, it follows that . In addition, as ℋ ∈_{ω}*SOC*(*X*,*τ*,_{ω}*A*), we have ∪̃{*H*:*H*∈ ℋ} = 1, and thus (∪̃{_{A}*H*:*H*∈ ℋ}) (*a*) = ∪{*H*(*a*) :*H*∈ ℋ} =*X*. It follows that .2. Let

*x*∈*X*. Then,*a*∈_{x}*SP*(*X*,*A*). As ℋ is soft locally finite in (*X*,*τ*,*A*), there is*F*∈*τ*such that*a*∊̃_{x}*F*, and {*H*∈ ℋ:*F*∩̃*H*≠ 0} is finite. Then,_{A}*x*∈*F*(*a*) ∈*τ*. Furthermore, if for_{a}*H*∈ ℋ, we have*F*(*a*) ∩*H*(*a*) ≠ ∅︀, then*F*∩̃*H*≠ 0. It follows that {_{A}*H*∈ ℋ:*F*(*a*) ∩*H*(*a*) ≠ ∅︀} is finite. Therefore, is locally finite in (*X*,*τ*)._{a}3. Let , where

*H*∈ ℋ. As , there is such that*H*⊂̃*a*; thus,_{U}*H*(*a*) ⊆*U*. This shows that .### Sufficiency

We assume that (

*X*,*τ*) is_{a}*ω*^{*}-paracompact. Let ℋ ∈*SOC*(*X*,*τ*,_{ω}*A*). Then, {*H*(*a*) :*H*∈ ℋ} ∈*OC*(*X*, (*τ*)_{ω}) =_{a}*OC*(*X*, (*τ*)_{a}). As (_{ω}*X*,*τ*) is_{a}*ω*^{*}-paracompact, there is such that is locally finite in (*X*,*τ*) and . Let ._{a}**Claim**(1) ℳ ∈

*SOC*(*X*,*τ*,_{ω}*A*).(2) ℳ is soft locally finite in (

*X*,*τ*,*A*).(3) ℳ ⪯̃ ℋ.

### Proof of Claim

(1) As , we have ℳ ⊆

*τ*. In addition, as , we have , and thus . It follows that ℳ ∈_{ω}*SOC*(*X*,*τ*,_{ω}*A*).(2) Let

*a*∈_{x}*SP*(*X*,*A*). As is locally finite in (*X*,*τ*), there is_{a}*O*∈*τ*such that_{a}*x*∈*O*and { :*O*∩*V*≠ ∅︀} is finite. Then,*a*∊̃_{x}*a*∈_{O}*τ*. If*a*∈ ℳ, where and_{V}*a*∩̃_{O}*a*≠ 0_{V}, then (_{A}*a*∩̃_{O}*a*) (_{V}*a*) =*O*∩*V*≠ ∅︀. It follows that ℳ is soft locally finite in*SOC*(*X*,*τ*,*A*).(3) Let

*a*∈ ℳ, where . As , there is_{V}*H*∈ ℋ such that*V*⊆*H*(*a*), and thus*a*⊂̃_{V}*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 {ℑ:_{a}*a*∈*A*} be an indexed family of topologies on*X*. Then,$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft*ω*^{*}-paracompact if and only if (*X*, ℑ) is_{a}*ω*^{*}-paracompact for all*a*∈*A*.**Proof. Necessity**We assume that

$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft*ω*^{*}-paracompact. Let*b*∈*A*. To show that (*X*, ℑ) is_{b}*ω*^{*}-paracompact, let . We set . Then,$\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega},A\right)$ . By Proposition 2.10,$\underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega}={\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega$ , and thus$\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$ . As$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft*ω*^{*}-paracompact, there is$\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$ such that ℳ is soft locally finite in$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ and ℳ ⪯̃ ℋ. By Proposition 2.10,$\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega},A\right)$ . Let ℳ= {_{b}*M*(*b*) :*M*∈ ℳ}.**Claim**(1) ℳ

∈_{b}*OC*(*X*, (ℑ)_{b})._{ω}(2) ℳ

is locally finite in (_{b}*X*, ℑ)._{b}(3) .

### Proof of Claim

1. As

$\mathcal{M}\subseteq {\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega}$ , by the definition of ℑ, we have ℳ_{b}⊆ (ℑ_{b})_{b}. As_{ω}$\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega},A\right)$ , we have ∪̃{*M*:*M*∈ ℳ} = 1; hence, ∪̃{_{A}*M*:*M*∈ ℳ})(*b*) = ∪{*M*(*b*) :*M*∈ ℳ} =*X*. Therefore, ℳ∈_{b}*OC*(*X*, (ℑ)_{b})._{ω}2. Let

*x*∈*X*. Then,*b*∈_{x}*SP*(*X*,*A*). As ℳ is soft locally finite in$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ , there is$F\in {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$ such that*b*∊̃_{x}*F*, and {*M*∈ ℳ:*F*∩̃*M*≠ 0} is finite. Thus,_{A}*x*∈*F*(*b*) ∈ ℑ. If_{b}*M*∈ ℳ is such that*F*(*b*) ∩*M*(*b*) ≠ ∅︀, then*F*∩̃*M*≠ 0. It follows that ℳ_{A}is locally finite in (_{b}*X*, ℑ)._{b}3. Let

*M*(*b*) ∈ ℳ− {∅︀}, where_{b}*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*=*b*and_{U}*H*(*b*) =*U*. It follows that .### Sufficiency

We assume that (

*X*, ℑ) is_{a}*ω*^{*}-paracompact for all*a*∈*A*. Let ℬ = {*a*:_{Y}*a*∈*A*and*Y*∈ (ℑ)_{a}}. Then, ℬ is a soft base of_{ω}$\underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega}={\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega$ . We apply Proposition 3.9. Let$\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$ with ℋ ⊆ ℬ. For each*a*∈*A*, let ℋ= {_{a}*Y*⊆*X*:*a*∈ ℋ}. Then, for all_{Y}*a*∈*A*, we have ℋ∈_{a}*OC*(*X*, (ℑ)_{a}_{ω}_{,}), where (*X*, ℑ) is_{a}*ω*^{*}-paracompact. Thus, there is ℳ∈_{a}*OC*(*X*, (ℑ)_{a}) such that ℳ_{ω}is locally finite in (_{a}*X*, ℑ) and ℳ_{a}⪯ ℋ_{a}. Let ._{a}**Claim**(1)

$\mathcal{G}\in SOC\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$ .(2) is soft locally finite in

$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ .(3) .

### Proof of Claim

1. For all

*a*∈*A*, we have ℳ⊆ (ℑ_{a})_{a}, and therefore_{ω}$\mathcal{G}\subseteq {\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega}={\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega}$ . For all*a*∈*A*, we have ℳ∈_{a}*OC*(*X*, (ℑ)_{a}), and thus (∪̃{_{ω}*a*:_{Y}*a*∈*A*and*Y*∈ ℳ}) (_{a}*a*) = ∪{*Y*:*Y*∈ ℳ} =_{a}*X*. Therefore, ∪̃{*a*:_{Y}*a*∈*A*and*Y*∈ ℳ} = 1_{a}. It follows that_{A}$\mathcal{G}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$ .2. Let

*b*∈_{x}*SP*(*X*,*A*). As ℳis locally finite in (_{a}*X*, ℑ), there is_{b}*O*∈ ℑsuch that_{b}*x*∈*O*, and {*Y*:*O*∩*Y*≠ ∅︀} is finite. We have${b}_{x}\in {b}_{O}\in {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$ . If*b*∩̃_{O}*a*≠ 0_{Y}, then_{A}*a*=*b*and*O*∩*Y*≠ ∅︀. This shows that {*a*:_{Y}*a*∈*A*and*Y*∈ ℳ, and_{a}*b*∩̃_{O}*a*≠ 0_{Y}} is finite. It follows that is soft locally finite in_{A}$\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ .3. Let with

*a*∈*A*and*Y*∈ ℳ. As ℳ_{a}⪯ ℋ_{a}, there is_{a}*Z*∈ ℋ, where_{a}*a*∈ ℋ, such that_{Z}*Y*⊆*Z*. Therefore,*a*⊂̃_{Y}*a*. It follows that ._{Z}

- 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*,*τ*,_{ω}*A*). Then, ℋ ∈*SOC*(*X*,*τ*,_{ω}*A*). As (*X*,*τ*,*A*) is soft*ω*^{*}-paracompact, there is ℳ ∈*SOC*(*X*,*τ*,_{ω}*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*,*τ*,_{ω}*A*) is soft paracompact.**Proof**We assume that (

*X*,*τ*,*A*) is soft*ω*^{*}-paracompact. Let ℋ ∈*SOC*(*X*,*τ*,_{ω}*A*). As (*X*,*τ*,*A*) is soft*ω*^{*}-paracompact, there is ℳ ∈*SOC*(*X*,*τ*,_{ω}*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*,*τ*,_{ω}*A*). It follows that (*X*,*τ*,_{ω}*A*) is soft paracompact.### Definition 4.3

An STS (

*X*,*τ*,*A*) is called strongly soft locally finite if for every*b*∈_{x}*SP*(*X*,*A*), there exists*H*∈*τ*such that*Supp*(*H*) is finite,*b*∊̃_{x}*H*, and*H*(*a*) is finite for all*a*∈*A*.### 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*b*∈_{x}*SP*(*X*,*A*). As (*X*,*τ*,*A*) is strongly soft locally finite, there exists*G*∈*τ*such that*Supp*(*G*) is finite,*b*∊̃_{x}*G*, and*G*(*a*) is finite for all*a*∈*A*. It is not difficult to verify that {*a*:_{x}*a*∊̃_{x}*G*} is finite. Thus, {*H*∈ ℋ:*G*∩̃*H*≠ 0} is finite. It follows that ℋ is soft locally finite in (_{A}*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*,*τ*,_{ω}*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*,*τ*,_{ω}*A*) is a discrete STS. Thus, ℳ ∈*SOC*(*X*,*τ*,_{ω}*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*,*τ*,_{ω}*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*,*τ*,_{ω}*A*) is a discrete STS, and thus ℋ ∈*SOC*(*X*,*τ*,_{ω}*A*). As (*X*,*τ*,*A*) is soft*ω*^{*}-paracompact, there is ℳ ∈*SOC*(*X*,*τ*,_{ω}*A*) such that ℳ is soft locally finite in (*X*,*τ*,*A*),ℳ ⪯̃ ℋ, and for every*M*∈ ℳ, we have*M*≠ 0. For every_{A}*M*∈ ℳ, there is*a*_{x}_{M}such that*M*⊂̃*a*_{x}_{M}, and thus*M*=*a*_{x}_{M}. Therefore, ℳ ⊆*SP*(*X*,*A*). As ℳ ∈*SOC*(*X*,*τ*,_{ω}*A*), we have ℳ =*SP*(*X*,*A*). As ℳ is soft locally finite in (*X*,*τ*,*A*), for every*a*∈_{x}*SP*(*X*,*A*), there is*G*∈*τ*such that*a*∊̃_{x}*G*, and {*M*∈ ℳ:*G*∩̃*M*≠ 0} = {_{A}*b*∈_{y}*SP*(*X*,*A*):*G*∩̃*b*≠ 0_{y}} is finite; thus,_{A}*Supp*(*G*) is finite, and for every*b*∈*A*,*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*,*τ*,_{ω}*A*) is not soft regular.**Proof**We assume toward a contradiction that (

*X*,*τ*,_{ω}*A*) is soft regular. As (*X*,*τ*,*A*) is soft separable, there is*G*∈*SCSS*(*X*,*A*) such that 1=_{A}*Cl*(_{τ}*G*). As (*X*,*τ*,*A*) is strongly soft anti-locally countable, we have 1∉_{A}*SCSS*(*X*,*A*); thus,*a*∊̃ 1_{x}−_{A}*G*. As (*X*,*τ*,_{ω}*A*) is soft regular and*G*is soft closed in (*X*,*τ*,_{ω}*A*), there exist*U*∈*τ*,_{ω}*V*∈*τ*and*H*∈*CSS*(*X*,*A*) such that*G*⊂̃*U*,*a*∊̃_{x}*V*−*H*, and*U*∩̃ (*V*−*H*) = 0. As 1_{A}=_{A}*Cl*(_{τ}*G*) and*a*∊̃_{x}*V*∈*τ*, there is*b*∊̃_{y}*G*∩̃*V*. As*b*∊̃_{y}*G*⊂̃*U*, there exist*W*∈*τ*and*M*∈*CSS*(*X*,*A*) such that*b*∊̃_{y}*W*−*M*⊂̃*U*. As*b*∊̃_{y}*U*∩̃*W*∈*τ*, we have*U*∩̃*W*∈*τ*− {0}. As (_{A}*X*,*τ*,*A*) is soft anti-locally countable, we have*U*∩̃*W*∉*CSS*(*X*,*A*). As*U*∩̃*W*−*H*∩̃*G*⊂̃*U*∩̃ (*V*−*H*) = 0, we have_{A}*U*∩̃*W*⊂̃*H*∩̃*G*, and thus*U*∩̃*W*∈*CSS*(*X*,*A*), which is a contradiction.### Theorem 4.8

Every soft anti-locally countable soft

*T*_{2}soft paracompact soft separable STS is not soft*ω*^{*}-paracompact.**Proof**We assume that (

*X*,*τ*,*A*) is soft anti-locally countable, soft*T*_{2}, 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*,*τ*,_{ω}*A*) is soft paracompact. As (*X*,*τ*,*A*) is soft*T*_{2}, it is clear that (*X*,*τ*,_{ω}*A*) is soft*T*_{2}. As (*X*,*τ*,_{ω}*A*) is soft*T*_{2}and soft paracompact, by Theorem 5.8 in [37], (*X*,*τ*,_{ω}*A*) is soft regular. However, as (*X*,*τ*,*A*) is soft anti-locally countable and soft separable, by Theorem 4.1, (*X*,*τ*,_{ω}*A*) is not soft regular, which is a contradiction.### Example 4.9

Let

*A*= {*a*},*X*= ℝ, and*τ*= {*F*∈*SS*(*X*,*A*) :*F*(*a*) belongs to the usual topology on ℝ}. Then, clearly, (*X*,*τ*,*A*) is soft anti-locally countable, soft*T*_{2}, 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*= {*F*∈*SS*(*X*,*A*) :*F*(*a*) ∈ ℑ}. Clearly, (*X*,*τ*,_{ω}*A*) is a discrete STS. However, by Theorem 4.6, (*X*,*τ*,_{ω}*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.

- References
- Al Ghour, S, and Bin-Saadon, A (2019). On some generated soft topological spaces and soft homogeneity. Heliyon.
*5*. article no. e02061 - Al Ghour, S, and Hamed, W (2020). On two classes of soft sets in soft topological spaces. Symmetry.
*12*. article no. 265 - Molodtsov, D (1999). Soft set theory: first results. Computers and Mathematics with Applications.
*37*, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5 - Shabir, M, and Naz, M (2011). On soft topological spaces. Computers and Mathematics with Applications.
*61*, 1786-1799. https://doi.org/10.1016/j.camwa.2011.02.006 - Oguz, G (2020). Soft topological transformation groups. Mathematics.
*8*. article no. 1545 - Cetkin, V, Guner, E, and Aygun, H (2020). On 2S-metric spaces. Soft Computing.
*24*, 12731-12742. https://doi.org/10.1007/s00500-020-05134-w - El-Shafei, ME, and Al-shami, TM (2020). Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem. Computational and Applied Mathematics.
*39*. article no. 138 - Alcantud, JCR (2020). Soft open bases and a novel construction of soft topologies from bases for topologies. Mathematics.
*8*. article no. 672 - Bahredar, AA, and Kouhestani, N (2020). On ɛ-soft topological semigroups. Soft Computing.
*24*, 7035-7046. https://doi.org/10.1007/s00500-020-04826-7 - Min, WK (2020). On soft ω-structures defined by soft sets. International Journal of Fuzzy Logic and Intelligent Systems.
*20*, 119-123. http://doi.org/10.5391/IJFIS.2020.20.2.119 - Al-shami, TM, and El-Shafei, ME (2020). Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone. Soft Computing.
*24*, 5377-5387. https://doi.org/10.1007/s00500-019-04295-7 - Al-shami, TM, and El-Shafei, ME (2020). T-soft equality relation. Turkish Journal of Mathematics.
*44*, 1427-1441. https://doi.org/10.3906/mat-2005-117 - Al-shami, TM, and Kocinac, L (2020). Nearly soft Menger spaces. Journal of Mathematics.
*2020*. article no. 3807418 - Al-shami, TM, Kocinac, L, and Asaad, BA (2020). Sum of soft topological spaces. Mathematics.
*8*. article no. 990 - Al-shami, TM, and El-Shafei, ME (2020). Two new forms of ordered soft separation axioms. Demonstratio Mathematica.
*53*, 8-26. https://doi.org/10.1515/dema-2020-0002 - Al-shami, TM, and El-Shafei, ME (2019). On supra soft topological ordered spaces. Arab Journal of Basic and Applied Sciences.
*26*, 433-445. https://doi.org/10.1080/25765299.2019.1664101 - Kiruthika, M, and Thangavelu, P (2019). A link between topology and soft topology. Hacettepe Journal of Mathematics and Statistics.
*48*, 800-804. https://doi.org/10.15672/HJMS.2018.551 - Al-Omari, A (2019). Soft topology in ideal topological spaces. Hacettepe Journal of Mathematics and Statistics.
*48*, 1277-1285. https://doi.org/10.15672/HJMS.2018.557 - Al-shami, TM, El-Shafei, ME, and Abo-Elhamayel, M (2019). On soft topological ordered spaces. Journal of King Saud University-Science.
*31*, 556-566. https://doi.org/10.1016/j.jksus.2018.06.005 - Polat, NC, Yaylali, G, and Tanay, B (2019). Some results on soft element and soft topological space. Mathematical Methods in the Applied Sciences.
*42*, 5607-5614. https://doi.org/10.1002/mma.5778 - Abbas, M, Murtaza, G, and Romaguera, S (2019). Remarks on fixed point theory in soft metric type spaces. Filomat.
*33*, 5531-5541. https://doi.org/10.2298/FIL1917531A - Al-shami, TM, El-Shafei, ME, and Abo-Elhamayel, M (2018). Almost soft compact and approximately soft Lindelof spaces. Journal of Taibah University for Science.
*12*, 620-630. https://doi.org/10.1080/16583655.2018.1513701 - Polat, NC, Yaylali, G, and Tanay, B (2018). A new approach for soft semi-topological groups based on soft element. Filomat.
*32*, 5743-5751. https://doi.org/10.2298/FIL1816743P - El-Shafei, ME, Abo-Elhamayel, M, and Al-shami, TM (2018). Partial soft separation axioms and soft compact spaces. Filomat.
*32*, 4755-4771. https://doi.org/10.2298/FIL1813755E - Ozturk, TY, Aras, CG, and Yolcu, A (2018). Soft bigeneralized topological spaces. Filomat.
*32*, 5679-5690. https://doi.org/10.2298/FIL1816679O - Tahat, MK, Sidky, F, and Abo-Elhamayel, M (2018). Soft topological soft groups and soft rings. Soft Computing.
*22*, 7143-7156. https://doi.org/10.1007/s00500-018-3026-z - Kandemir, MB (2018). The concept of σ-algebraic soft set. Soft Computing.
*22*, 4353-4360. https://doi.org/10.1007/s00500-017-2901-3 - Al-Saadi, HS, and Min, WK (2017). On soft generalized closed sets in a soft topological space with a soft weak structure. International Journal of Fuzzy Logic and Intelligent Systems.
*17*, 323-328. https://doi.org/10.5391/IJFIS.2017.17.4.323 - Hdeib, H (1982). ω-closed mappings. Revista Colombiana de Matematicas.
*16*, 65-78. - Al Ghour, S, and Irshedat, B (2020). On θ continuity. Heliyon.
*6*. article no. e03349 - Roy, B (2019). On nearly Lindelof spaces via generalized topology. Proyecciones (Antofagasta).
*38*, 49-57. https://doi.org/10.4067/S0716-09172019000100049 - Agarwal, P, and Goel, CK (2019). Delineation of Ω bitopological spaces. Proceedings of the Jangjeon Mathematical Society.
*22*, 507-516. - Al Ghour, S (2018). Theorems on strong paracompactness of product spaces. Mathematical Notes.
*103*, 54-58. https://doi.org/10.1134/S0001434618010066 - Al Ghour, S, and Mansur, K (2018). Between open sets and semiopen sets. Universitas Scientiarum.
*23*, 9-20. https://doi.org/10.11144/Javeriana.SC23-1.bosa - Al Ghour, S, and Irshedat, B (2017). The topology of θ-open sets. Filomat.
*31*, 5369-5377. https://doi.org/10.2298/FIL1716369A - Al Ghour, S (2002). On the product of meta Lindelof spaces. Far East Journal of Mathematical Sciences.
*4*, 383-388. - Rong, W, and Lin, F (2013). Soft connected spaces and soft paracompact spaces. International Journal of Applied Mathematics and Statistics.
*51*, 667-681. - Steen, LA, and Seebach, J (1970). Counterexamples in Topology. New York, NY: Holt, Rinehart and Winston

- 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