### The Fixed Point Property of the Smallest Open Neighborhood of the n-dimensional Khalimsky Topological Space

#### Abstract

The paper aims to propose the fixed point property({\it FPP} for short) of a Khalimsky ($K$-, for short) retract.

Let $(X, \kappa_X^n)$ be an $n$D Khalimsky topological space induced by the $n$D Khalimsky space denoted by $({\bf Z}^n, \kappa^n)$.

Although not every Khalimsky space $(X, \kappa_X^n)$ has the {\it FPP}, we prove that for any point $x \in {\bf Z}^n$ the smallest open $K$-topological neighborhood of $x$, denoted by $SN_K(x) \subset (X, \kappa_X^n)$, has the {\it FPP}.

Besides, the present paper studies the almost fixed point property ({\it AFPP}, for brevity) of a $K$-topological space.

Let $(X, \kappa_X^n)$ be an $n$D Khalimsky topological space induced by the $n$D Khalimsky space denoted by $({\bf Z}^n, \kappa^n)$.

Although not every Khalimsky space $(X, \kappa_X^n)$ has the {\it FPP}, we prove that for any point $x \in {\bf Z}^n$ the smallest open $K$-topological neighborhood of $x$, denoted by $SN_K(x) \subset (X, \kappa_X^n)$, has the {\it FPP}.

Besides, the present paper studies the almost fixed point property ({\it AFPP}, for brevity) of a $K$-topological space.

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