The fixed point property of the infinite $K$-sphere in the category $Top(({\mathbb Z}^2)^\ast)$

Abstract

In this paper the Alexandroff one point compactification of the $2$-dimensional Khalimsky ($K$-, for brevity) plane ({\it resp.} the $1$-dimensional Khalimsky line) is
 called the infinite $K$-sphere ({\it resp.} the infinite $K$-circle).
  The present paper firstly proves the non-fixed point property (non-{\it FPP}, for short) of both the infinite $K$-sphere and the infinite $K$-circle.
  Let us mathematically denote by $(({\mathbb Z}^2)^\ast, (\kappa^2)^\ast)$ ({\it resp.} $({\mathbb Z}^\ast, \kappa^\ast))$
  the infinite $K$-sphere ({\it resp.}  the infinite $K$-circle) (see Theorem 4.1).
  Next, we address the following query: Under what category does the infinite $K$-sphere (or the infinite $K$-circle) have the {\it FPP}\,?
   Motivated by this query, we prove that the infinite $K$-sphere has the {\it FPP} in the category $Top(({\mathbb Z}^2)^\ast)$, where $Top(({\mathbb Z}^2)^\ast)$ means the category
   with the infinite $K$-sphere as an object and the set of all continuous self-maps $f$ of the infinite $K$-sphere such that
$\vert\,f(({\mathbb Z}^2)^\ast)\,\vert = \aleph_0$ with $\ast \in f(({\mathbb Z}^2)^\ast)$ or constant maps as morphisms (see Definition 1).
       We also prove that the infinite $K$-circle has the similar feature (see Corollary 4.6).
  In view of these results, we can recognize that the {\it FPP} problems of the infinite $K$-sphere and the infinite $K$-circle are
   quite different from the {\it FPP} of the Hausdroff one point compactification of the $n$-dimensional Euclidean topological space, $n \in \{1, 2\}$.
       \end{abstract}