Closed graph property and Khalimsky spaces

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 $\hat{I}{»}_1,\dots ,\hat{I}{»}_n$ such that $f$ is a constant map with value $(2\hat{I}{»}_1,\cdots ,2\hat{I}{»}_n)$. 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.

5 pages