# Some Variants of Normal Čech Closure Spaces via Canonically Closed Sets

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1.**

- 1.
- normal if for every two disjoint closed sets $A=cl\left(A\right)$ and $B=cl\left(B\right)$ there exist disjoint open sets U and V containing $cl\left(A\right)$ and $cl\left(B\right)$ respectively.
- 2.
- almost normal if for every two disjoint closed sets $cl\left(A\right)=A$ and $cl\left(B\right)=B$ out of which one is canonically closed there exist disjoint open sets U and V containing $cl\left(A\right)$ and $cl\left(B\right)$ respectively.
- 3.
- weakly normal if for every two disjoint closed sets $cl\left(A\right)=A$ and $cl\left(B\right)=B$ there exists an open set U such that $A\subseteq U$ and $int\left(cl\right(U\left)\right)\cap B=\varnothing $.

**Remark**

**1.**

**Lemma**

**1.**

**Theorem**

**1.**

## 2. Variants of Normal Čech Closure Space

**Definition**

**2.**

**Example**

**1.**

**Definition**

**3.**

**Example**

**2.**

- Type-I: A is finite in X.
- Type-II: A is infinite in Y such that $p\notin A$ and $q\notin A$.
- Type-III: $(Y-A)$ is finite and A contains either p or q.
- Type-IV: $(Y-A)$ is finite and A contains both p and q.

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Definition**

**4.**

**Definition**

**5.**

- ${T}_{1}$ if for two distinct points x and y, we have $x\notin cl\left(\right\{y\left\}\right)$ and $y\notin cl\left(\right\{x\left\}\right)$.
- ${T}_{2}$ if any two distinct points x and y are separated.

**Remark**

**2.**

**Definition**

**6.**

**Theorem**

**4.**

**Proof.**

**Definition**

**7.**

**Definition**

**8.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Example**

**11.**

**Definition**

**9.**

**Definition**

**10.**

**Example**

**12.**

**Theorem**

**9.**

**Proof.**

**Example**

**13.**

**Example**

**14.**

**Theorem**

**10.**

- 1.
- $(X,cl)$ is normal.
- 2.
- $(X,cl)$ is π-normal.
- 3.
- $(X,cl)$ is weakly π-normal.
- 4.
- $(X,cl)$ is κ-normal.
- 5.
- $(X,cl)$ is almost normal.

## 3. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Gupta, R.; Das, A.K.
Some Variants of Normal Čech Closure Spaces via Canonically Closed Sets. *Mathematics* **2021**, *9*, 1225.
https://doi.org/10.3390/math9111225

**AMA Style**

Gupta R, Das AK.
Some Variants of Normal Čech Closure Spaces via Canonically Closed Sets. *Mathematics*. 2021; 9(11):1225.
https://doi.org/10.3390/math9111225

**Chicago/Turabian Style**

Gupta, Ria, and Ananga Kumar Das.
2021. "Some Variants of Normal Čech Closure Spaces via Canonically Closed Sets" *Mathematics* 9, no. 11: 1225.
https://doi.org/10.3390/math9111225