Good Coverings of Proximal Alexandrov Spaces. Path Cycles in the Extension of the Mitsuishi-Yamaguchi good covering and Jordan Curve Theorems

TL;DR

This paper extends classical topological theorems to proximal Alexandrov spaces using path cycles, introducing new forms of good covers and applying these to analyze the persistence of shapes in video frames.

Contribution

It introduces proximal path cycles and extends the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan Curve Theorem to Alexandrov spaces with proximity relations.

Findings

Extended Mitsuishi-Yamaguchi Good Covering Theorem

Extended Jordan Curve Theorem for proximal Alexandrov spaces

Application to persistence of shapes in video frames

Abstract

This paper introduces proximal path 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 as well as extension of the Jordan curve theorem. In this work, a {\bf path cycle} is a sequence of maps $h_1,\dots,h_i,\dots,h_{n-1}\mbox{mod}\ n$ in which $h_i:[0,1]\to X$ and $h_i(1) = h_{i+1}(0)$ provide the structure of a path-connected cycle that has no end path. An application of these results is also given for the persistence of proximal video frame shapes that appear in path cycles.