Topologies on Zn that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

Mathematics

Saeid Jafari¹  Sang-Eon Han²  Jeong Min Kang³ 

[1] College of Vestsjaelland South Herrestraede 114200 Slagelse, Denmark;Department of Mathematics Education, Institute of Pure and Applied Mathematics Jeonbuk National University, Jeonju-City 54896, Jeonbuk, Korea;Mathematics, School of Liberal, Arts Education, University of Seoul, Seoul 02504, Korea;

关键词: khalimsky topology;    quasi-discrete (clopen or pseudo-discrete);    t12-separation axiom;    alexandroff topology;    digital topology;   

DOI  :  10.3390/math7111072

来源: DOAJ

【 摘 要 】

The present paper deals with two types of topologies on the set of integers, $\mathbb{Z}$ : a quasi-discrete topology and a topology satisfying the $T_{\frac{1}{2}}$ -separation axiom. Furthermore, for each $n\in \mathbb{N}$ , we develop countably many topologies on ${\mathbb{Z}}^n$ which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on ${\mathbb{Z}}^n$ , many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on.

【 授权许可】

Unknown