Topologies on Zn that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space - Sang-Eon Han; Saeid Jafari; Jeong Min Kang (Mathematics, 2019)
The present paper deals with two types of topologies on the set of integers, <semantics> Z </semantics> : a quasi-discrete topology and a topology satisfying the <semantics> T 1 2 </semantics> -separation axiom. Furthermore, for each <semantics> n ∈ N </semantics> , we develop countably many topologies on <semantics> Z n </semantics> which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on <semantics> Z n </semantics> , 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.