# 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

## References

- Čech, E. Topological Spaces (Revised edition by Zdeněk Frolík and Miroslav Katětov); Publishing House of The Czechoslovak Academy of Sciences: Prague, Czech Republic, 1966. [Google Scholar]
- Császár, Á. Generalized Topology, Generalized Continuity. Acta Math. Hung.
**2002**, 96, 351–357. [Google Scholar] [CrossRef] - Császár, Á. Weak structures. Acta Math. Hung.
**2011**, 131, 193–195. [Google Scholar] [CrossRef] - Das, A.K. A note on weak structures due to Császár. Bul. Acad. Ştiinţe Mold. Mat.
**2015**, 78, 114–116. [Google Scholar] - Min, W.K. Some Results on generalized topological spaces and generalized systems. Acta Math. Hung.
**2005**, 108, 171–181. [Google Scholar] [CrossRef] - Šlapal, J. Closure operations for digital topology. Theor. Comput. Sci.
**2003**, 305, 457–471. [Google Scholar] [CrossRef][Green Version] - Khalimsky, E.D.; Kopperman, R.; Meyer, P.R. Computer graphics and connected topologies on finite ordered sets. Topol. Appl.
**1990**, 36, 1–17. [Google Scholar] [CrossRef][Green Version] - Rosenfeld, A.; Kak, A.C. Digital Picture Processing; Academic Press: New York, NY, USA, 1976. [Google Scholar]
- Rosenfeld, A. Digital topology. Amer. Math. Mon.
**1979**, 86, 621–630. [Google Scholar] [CrossRef] - Kuratowski, C. Sur l operation de l Analysis Situs. Fundam. Math.
**1922**, 3, 182–199. [Google Scholar] [CrossRef] - Galton, A. A generalized topological view of motion in discrete space. Theoret. Comput. Sci.
**2003**, 305, 111–134. [Google Scholar] [CrossRef][Green Version] - Allam, A.A.; Bakeir, M.Y.; Abo-Tabl, E.A. New approach for closure spaces by relations. Acta Math. Acad. Paedagog. Nyregyhziensis
**2006**, 22, 285–304. [Google Scholar] - Allam, A.A.; Bakeir, M.Y.; Abo-Tabl, E.A. Some methods for generating topologies by relations. Bull. Malays. Math. Sci. Soc.
**2008**, 31, 1–11. [Google Scholar] - Liu, G. Closures and topological closures in quasi-discrete closure spaces. Appl. Math. Lett.
**2010**, 23, 772–776. [Google Scholar] [CrossRef][Green Version] - Šlapal, J.; Pfaltz, J.L. Closure operators associated with networks. Hacet. J. Math. Stat.
**2017**, 46, 91–101. [Google Scholar] - Stadler, B.M.R.; Stadler, P.F. Higher separation axioms in generalized closure spaces. Comment. Math. Prace Mat.
**2003**, 43, 257–273. [Google Scholar] - Gupta, R.; Das, A.K. New separation axioms on closure spaces generated by relations. Proc. Jangjeon Math. Soc.
**2018**, 21, 23–31. [Google Scholar] - Das, A.K.; Bhat, P.; Tartir, J.K. On a simultaneous generalization of β-normality and almost β-normality. Filomat
**2017**, 31, 425–430. [Google Scholar] [CrossRef] - Das, A.K. Simultaneous generalizations of regularity and normality. Eur. J. Pure Appl. Math.
**2011**, 4, 34–41. [Google Scholar] - Das, A.K.; Raina, S.S. On Relative β-Normality. Acta Math. Hung.
**2020**, 160, 468–477. [Google Scholar] [CrossRef] - Kalantan, L.; Saeed, M.M. L-normality. Topol. Proc.
**2017**, 50, 141–149. [Google Scholar] - Singal, M.K.; Singal, A.R. Mildly normal spaces. Kyungpook Math. J.
**1973**, 13, 27–31. [Google Scholar] - Das, A.K.; Gupta, R. On Some Variants of Normality in Čech Closure Spaces; Unpublished work.
- Gupta, R.; Das, A.K. On β-Normal Čech Closure Spaces; Unpublished work.
- Andrijević, D.; Jelić, M.; Mršević, M. Some properties of hyperspaces of Čech closure spaces with Vietoris-like topologies. Filomat
**2010**, 24, 53–61. [Google Scholar] [CrossRef] - Andrijević, D.; Jelić, M.; Mršević, M. On function spaces topologies in the setting of Čech closure spaces. Topology Appl.
**2011**, 158, 1390–1395. [Google Scholar] [CrossRef][Green Version] - Rieser, A. Čech closure spaces: A unified framework for discrete and continuous homotopy. Topology Appl.
**2021**, 296, 107613. [Google Scholar] [CrossRef] - Qiao, J. On the reflective and coreflective subcategory of stratified L-Čech closure spaces. Fuzzy Sets Syst.
**2020**, 390, 105–117. [Google Scholar] [CrossRef] - Perfilieva, I.; Ramadan, A.A.; Elkordy, E.H. Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces. Mathematics
**2020**, 8, 1274. [Google Scholar] [CrossRef] - Močkoř, J. Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces. Axioms
**2020**, 9, 63. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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