The Fixed Point Property of the Infinite M-Sphere

Abstract

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space ( Z 2 , γ ) . This compactification is called the infinite M-topological sphere and denoted by ( ( Z 2 ) ∗ , γ ∗ ) , where ( Z 2 ) ∗ : = Z 2 ∪ { ∗ } , ∗ ∉ Z 2 and γ ∗ is the topology for ( Z 2 ) ∗ induced by the topology γ on Z 2 . With the topological space ( ( Z 2 ) ∗ , γ ∗ ) , since any open set containing the point “ ∗ ” has the cardinality ℵ 0 , we call ( ( Z 2 ) ∗ , γ ∗ ) 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 ( ( Z 2 ) ∗ , γ ∗ ) have the fixed point property (FPP, for short)? The present paper proves that ( ( Z 2 ) ∗ , γ ∗ ) has the FPP in the category M o p ( γ ∗ ) whose object is the only ( ( Z 2 ) ∗ , γ ∗ ) and morphisms are all continuous self-maps g of ( ( Z 2 ) ∗ , γ ∗ ) such that | g ( ( Z 2 ) ∗ ) | = ℵ 0 with ∗ ∈ g ( ( Z 2 ) ∗ ) or g ( ( Z 2 ) ∗ ) is a singleton. Since ( ( Z 2 ) ∗ , γ ∗ ) can be a model for a digital sphere derived from the M-topological space ( Z 2 , γ ) , it can play a crucial role in topology, digital geometry and applied sciences.