Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

Abstract

<abstract><p>The present paper intensively studies various properties of certain topologies on the set of integers $ {\mathbb Z} $ (resp. $ {\mathbb Z}^n $) which are either homeomorphic or not homeomorphic to the typical Khalimsky line topology (resp. $ n $-dimensional Khalimsky topology). This finding plays a crucial role in addressing some problems which remain open in the field of digital topology.</p></abstract>