The Fixed Point Property of the Infinite M-Sphere


Han S., ÖZÇAĞ S.

MATHEMATICS, cilt.8, sa.4, 2020 (SCI İndekslerine Giren Dergi) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 8 Konu: 4
  • Basım Tarihi: 2020
  • Doi Numarası: 10.3390/math8040599
  • Dergi Adı: MATHEMATICS

Özet

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space . This compactification is called the infinite M-topological sphere and denoted by ((Z2),gamma), where (Z2):=Z2{},is not an element of Z2 and gamma is the topology for (Z2) induced by the topology gamma on Z2. With the topological space ((Z2),gamma), since any open set containing the point "" has the cardinality aleph 0, we call ((Z2),gamma) 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 ((Z2),gamma) have the fixed point property (FPP, for short)? The present paper proves that ((Z2),gamma) has the FPP in the category Mop(gamma) whose object is the only ((Z2),gamma) and morphisms are all continuous self-maps g of ((Z2),gamma) such that |=aleph 0 with is an element of g((Z2)) or g((Z2)) is a singleton. Since ((Z2),gamma) can be a model for a digital sphere derived from the M-topological space , it can play a crucial role in topology, digital geometry and applied sciences.