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

• Peters, James Francis ^[1] ; Vergili, Tane ^[2]
  1. [1] University of Manitoba

     University of Manitoba

     Canadá

     
  2. [2] Karadeniz Technical University

     Karadeniz Technical University

     Turquía

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 24, Nº. 1, 2023, págs. 25-45
• Idioma: inglés
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• Resumen

  • 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.