# Caliber and Chain Conditions in Soft Topologies

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 2.1. Elements of (Crisp) Topology

## 2.2. Elements of Soft Set Theory and Soft Topology

- (1)
- $({F}_{1},E)\bigsqcup ({F}_{2},E)$ is the soft set $({F}_{3},E)\in S{S}_{E}\left(X\right)$ such that ${F}_{3}\left(a\right)={F}_{1}\left(a\right)\cup {F}_{2}\left(a\right)$ for each $a\in E$.This can also be written as $(({F}_{1},E)\bigsqcup ({F}_{2},E))\left(a\right)={F}_{1}\left(a\right)\cup {F}_{2}\left(a\right)$ when $a\in E$.
- (2)
- $({F}_{1},E)\sqcap ({F}_{2},E)$ is the soft set $({F}_{4},E)\in S{S}_{E}\left(X\right)$ such that ${F}_{4}\left(a\right)={F}_{1}\left(a\right)\cap {F}_{2}\left(a\right)$ for each $a\in E$.This can also be written as $(({F}_{1},E)\sqcap ({F}_{2},E))\left(a\right)={F}_{1}\left(a\right)\cap {F}_{2}\left(a\right)$ when $a\in E$.
- (3)
- $({F}_{1},E)\u2291({F}_{2},E)$ represents the property ${F}_{1}\left(a\right)\subseteq {F}_{2}\left(a\right)$ when $a\in E$.

**Remark**

**1.**

**Definition**

**1.**

- (1)
- $\Phi ,\tilde{X}\in \tau $;
- (2)
- The union of soft sets from τ is a member of τ;
- (3)
- The intersection of a finite number of soft sets from τ is a member of τ.

**Example**

**1.**

- $\left(i\right)$
- The indiscrete soft topology is ${\tau}_{id}=\{\mathrm{\Phi},\tilde{\mathrm{X}}\}$, and the discrete soft topology is ${\tau}_{d}=S{S}_{E}\left(X\right)$ [7].
- $\left(ii\right)$
- The cofinite soft topology was defined in [10] as$${\tau}_{c}=\left\{(F,E)\in S{S}_{E}\left(X\right):{(F,E)}^{c},thecomplementof(F,E),isafinitesoftset\right\}.$$

**Definition**

**2.**

**Remark**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

- $\left(\mathbf{i}\right)$
- soft continuous, when $(F,E)\in {\tau}^{\prime}$ implies ${h}_{\phi}^{-1}(F,E)\in \tau $;
- $\left(\mathbf{i}\mathbf{i}\right)$
- soft open when ${h}_{\phi}(F,E)\in {\tau}^{\prime}$ if $(F,E)\in \tau $;
- $\left(\mathbf{i}\mathbf{i}\mathbf{i}\right)$
- a soft homeomorphism when it is both soft continuous, soft open and bijective.

## 3. New Concepts in Soft Topology: Calibers and Chain Conditions

## 3.1. Axioms in Soft Topology

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

## 3.2. Axioms in Soft Topology Derived from Topological Conditions

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Remark**

**3.**

**Proposition**

**4.**

**Proof.**

## 4. Main Results

#### 4.1. The Role of Cardinality in FCC and CCC

**Proposition**

**5.**

**Proof.**

**Example**

**2.**

**Example**

**3.**

#### 4.2. Some Relationships among Properties

**Proposition**

**6.**

**Proof.**

**Remark**

**4.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Corollary**

**1.**

**Corollary**

**2.**

**Example**

**4.**

#### 4.3. A Construction of Soft Topologies That Satisfy CCC

**Corollary**

**3.**

**Proof.**

**Example**

**5.**

#### 4.4. Preservation of Properties by Mappings

**Proposition**

**9.**

**Proof.**

**Proposition**

**10.**

**Proof.**

## 5. Conclusions

- 1.
- If $\tau $ on X has caliber ${\aleph}_{0}$ (resp., caliber ${\aleph}_{1}$) then $\tau $ satisfies FCC (resp., CCC): cf., Proposition 6;
- 2.
- Suppose that E is finite.(2.1) Then when X is finite, FCC holds true trivially.(2.2) If this is not the case, but still there is a finite $\tau $-soft dense set, then the caliber of $\tau $ is ${\aleph}_{0}$ and in particular, FCC holds true as well.(2.3) Observe that statement (2.1) is a particular case of (2.2), since a finite X ensures that the absolute soft set $\tilde{X}$ is a finite $\tau $-soft dense set;
- 3.
- Suppose instead that E is countable.(3.1) Then when X is countable, CCC holds true: cf., Proposition 5.(3.2) If this is not the case, but still there is a countable $\tau $-soft dense set (i.e., $\tau $ is CS${}^{2}$D), then the caliber of $\tau $ is ${\aleph}_{1}$ (cf., Proposition 8) and in particular, CCC holds true as well.(3.3) Again, statement (3.1) is a particular case of (3.2): notice that a countable X ensures that the collection of all soft points is soft $\tau $-dense in X and of course it is countable, therefore $\tau $ satisfies SPCD and CS${}^{2}$D.

- 4.
- If $\tau $ has caliber ℵ then ${\Sigma}_{e}$ has caliber ℵ, for each $e\in E$: cf., Proposition 1;
- 5.
- Suppose that E is finite. Then, when each ${\Sigma}_{e}$ has caliber ${\aleph}_{0}$, $\tau $ has caliber ${\aleph}_{0}$ (v., Proposition 2) and, when each ${\Sigma}_{e}$ has caliber ${\aleph}_{1}$, $\tau $ has caliber ${\aleph}_{1}$ (v., Proposition 3);
- 6.
- If $\tau $ satisfies CCC then ${\Sigma}_{e}$ satisfies CCC, for each $e\in E$ (v., part 1 of Proposition 4);
- 7.
- Suppose that E is countable. Then when each ${\Sigma}_{e}$ satisfies CCC, $\tau $ satisfies CCC (v., part 2 of Proposition 4).

## Author Contributions

## Funding

## Institutional Review Board Statement:

## Informed Consent Statement:

## Data Availability Statement:

## Conflicts of Interest

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Alcantud, J.C.R.; Al-shami, T.M.; Azzam, A.A. Caliber and Chain Conditions in Soft Topologies. *Mathematics* **2021**, *9*, 2349.
https://doi.org/10.3390/math9192349

**AMA Style**

Alcantud JCR, Al-shami TM, Azzam AA. Caliber and Chain Conditions in Soft Topologies. *Mathematics*. 2021; 9(19):2349.
https://doi.org/10.3390/math9192349

**Chicago/Turabian Style**

Alcantud, José Carlos R., Tareq M. Al-shami, and A. A. Azzam. 2021. "Caliber and Chain Conditions in Soft Topologies" *Mathematics* 9, no. 19: 2349.
https://doi.org/10.3390/math9192349