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

2021/04/06 by J. F. Peters, Peters, J. F., T. Vergili +1 · 2 citations
Computer Science · Engineering · Mathematics · #54E05 #55P57 #Advanced Numerical Analysis Techniques #Algebraic Topology (math.AT) #Computational Geometry and Mesh Generation #Computer Graphics and Visualization Techniques #FOS: Mathematics #Geometric Topology (math.GT) #math.AT #math.GT #msc:54E05 #msc:55P57

paper · pdf · doi:10.48550/arxiv.2104.05601

22 pages, 7 figures

openalex publication_date 2021/04/06 · arxiv created 2022/01/19 · arxiv updated 2022/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 h₁,…,h_i,…,h_n-1mod n in which h_i:[0,1]→ 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.

Cited by

• Proximal and Descriptive Topological Rings: Structure and Properties
• Good Coverings of Proximal Alexandrov Spaces. Homotopic Cycles in Jordan Curve Theorem Extension

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