Remarks on topological spaces on Zⁿ which are related to the Khalimsky n-dimensional space

Abstract

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.

Keywords

• khalimsky topology
• t_&#189-separation axiom
• alexandroff topology
• quasi-discrete
• digital topology