Closed graph property and Khalimsky spaces

TL;DR

This paper characterizes self-maps with closed graphs on Khalimsky spaces, showing they are constant maps with specific values, and extends results to circles and spheres within this topological framework.

Contribution

It provides a complete characterization of maps with closed graphs on Khalimsky spaces, identifying them as constant maps with particular values.

Findings

Self-maps with closed graphs on $\mathcal{K}^n$ are constant maps with specific integer values.

On Khalimsky circle and sphere, all maps with closed graphs are constant.

The results are motivated by and supported through examples.

Abstract

In the following text for Khalimsky $n-$dimensional space $\mathcal{K}^n$ we show self--map $f:\mathcal{K}^n\to\mathcal{K}^n$ has closed graph if and only if there exist integers $\lambda_1,\ldots,\lambda_n$ such that $f$ is a constant map with value $(2\lambda_1,\cdots,2\lambda_n)$. We also show each self--map on Khalimsky circle and Khalimsky sphere which has closed graph is a constant map. The text is motivated by examples.