Good Coverings of Proximal Alexandrov Spaces. Homotopic Cycles in Jordan Curve Theorem Extension

TL;DR

This paper extends classical theorems like the Jordan curve theorem and Mitsuishi-Yamaguchi's good covering theorem to proximal Alexandrov spaces, introducing proximal homotopic cycles and new forms of Tanaka good covers.

Contribution

It introduces proximal homotopic cycles and extends key topological theorems to Alexandrov spaces with proximity relations, providing new tools for topological analysis.

Findings

Extended Mitsuishi-Yamaguchi Good Covering Theorem for proximal Alexandrov spaces

Developed new forms of Tanaka good covers in the context of proximity spaces

Extended the Jordan curve theorem to proximal Alexandrov spaces

Abstract

This paper introduces proximal homotopic cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation and extensions of the Jordan curve theorem. An application of these results is also given.