Glasnik Matematicki, Vol. 56, No. 1 (2021), 175-194.

AMIABLE AND ALMOST AMIABLE FIXED SETS. EXTENSION OF THE BROUWER FIXED POINT THEOREM

James F. Peters

Computational Intelligence Laboratory, University of Manitoba, WPG, MB, R3T 5V6, Canada,
and
Department of Mathematics, Faculty of Arts and Sciences, Adıyaman University, 02040 Adıyaman, Turkey
e-mail: james.peters3@umanitoba.ca

Abstract.   This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovič-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovič and Yu.M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan curve theorem and a fixed cell complex theorem, which is an extension of the Brouwer fixed point theorem.

2010 Mathematics Subject Classification.   54E05, 55R40, 68U05, 55M20

Key words and phrases.   Amiable, boundary region, CW space, descriptive fixed set, descriptive proximally continuous map, descriptive proximity, wide ribbon

https://doi.org/10.3336/gm.56.1.11

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