Good coverings of proximal Alexandrov spaces. Path cycles in the extension of the Mitsuishi-Yamaguchi good covering and Jordan Curve Theorems

Peters J. F., Vergili T.

APPLIED GENERAL TOPOLOGY, vol.24, no.1, pp.25-45, 2023 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 24 Issue: 1
  • Publication Date: 2023
  • Doi Number: 10.4995/agt.2023.17046
  • Journal Name: APPLIED GENERAL TOPOLOGY
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, Academic Search Premier, zbMATH, Directory of Open Access Journals, DIALNET
  • Page Numbers: pp.25-45
  • Karadeniz Technical University Affiliated: Yes

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 path cycle is a sequence of maps h1,...,hi,...,hn-1 mod n in which hi  : [ 0,1 ] → X and hi(1) = hi+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.