# The Fixed Point Property of the Infinite M-Sphere

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Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, Korea

^{2}

Department of Mathematics, Hacettepe University, 06800 Ankara, Turkey

^{*}

Author to whom correspondence should be addressed.

Received: 2 March 2020 / Revised: 31 March 2020 / Accepted: 9 April 2020 / Published: 15 April 2020

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space $({\mathbb{Z}}^{2},\gamma )$ . This compactification is called the infinite M-topological sphere and denoted by $({\left({\mathbb{Z}}^{2}\right)}^{\ast},{\gamma}^{\ast})$ , where ${\left({\mathbb{Z}}^{2}\right)}^{\ast}:={\mathbb{Z}}^{2}\cup \{\ast \},\ast \notin {\mathbb{Z}}^{2}$ and ${\gamma}^{\ast}$ is the topology for ${\left({\mathbb{Z}}^{2}\right)}^{\ast}$ induced by the topology $\gamma $ on ${\mathbb{Z}}^{2}$ . With the topological space $({\left({\mathbb{Z}}^{2}\right)}^{\ast},{\gamma}^{\ast})$ , since any open set containing the point $\u201c\ast \u201d$ has the cardinality ${\aleph}_{0}$ , we call $({\left({\mathbb{Z}}^{2}\right)}^{\ast},{\gamma}^{\ast})$ the infinite M-topological sphere. Indeed, in the fields of digital or computational topology or applied analysis, there is an unsolved problem as follows: Under what category does $({\left({\mathbb{Z}}^{2}\right)}^{\ast},{\gamma}^{\ast})$ have the fixed point property (FPP, for short)? The present paper proves that $({\left({\mathbb{Z}}^{2}\right)}^{\ast},{\gamma}^{\ast})$ has the FPP in the category $Mop\left({\gamma}^{\ast}\right)$ whose object is the only $({\left({\mathbb{Z}}^{2}\right)}^{\ast},{\gamma}^{\ast})$ and morphisms are all continuous self-maps g of $({\left({\mathbb{Z}}^{2}\right)}^{\ast},{\gamma}^{\ast})$ such that $|\phantom{\rule{0.166667em}{0ex}}g\left({\left({\mathbb{Z}}^{2}\right)}^{\ast}\right)\phantom{\rule{0.166667em}{0ex}}|={\aleph}_{0}$ with $\ast \in g\left({\left({\mathbb{Z}}^{2}\right)}^{\ast}\right)$ or $g\left({\left({\mathbb{Z}}^{2}\right)}^{\ast}\right)$ is a singleton. Since $({\left({\mathbb{Z}}^{2}\right)}^{\ast},{\gamma}^{\ast})$ can be a model for a digital sphere derived from the M-topological space $({\mathbb{Z}}^{2},\gamma )$ , it can play a crucial role in topology, digital geometry and applied sciences.
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*Keywords:*Alexandroff one point compactification; Marcus-Wyse topology; infinite

*M*-topological sphere; fixed point property

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**MDPI and ACS Style**

Han, S.-E.; Özçağ, S. The Fixed Point Property of the Infinite *M*-Sphere. *Mathematics* **2020**, *8*, 599.

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