Topologies on <em>Z</em>ⁿ that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological Space

Abstract

The present paper deals with two types of topologies on the set of integers, Z : a quasi-discrete topology and a topology satisfying the T 1 2 -separation axiom. Furthermore, for each n ∈ N , we develop countably many topologies on Z n which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on 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.

Keywords

• khalimsky topology
• quasi-discrete (clopen or pseudo-discrete)
• t12-separation axiom
• alexandroff topology
• digital topology