Strong homotopy in finite topological adjacency category
type: Item
instance of: scholarly article
title: Strong homotopy in finite topological adjacency category
author: Yanyan Li
published in: Topology and its Applications
publication date: 2021-08-27T00:00:00Z
review text: Motivated by the \(KA\)- (or \(A\)-, for brevity) contractibility of \(SC_{KA}^{n,4}\) (see the first line of the proof of Theorem 5.1 of [\textit{S. E. Han}, Ukr. Math. J. 67, No. 8, 1264--1276 (2016; Zbl 1377.68280)]), the present paper develops the so-called (strong) \(SA\)-homotopy for finite topological spaces, i.e., Alexandroff spaces (see Definition 3.2 of the present paper). Since \(SA\)-homotopy is very rigid compared to \(KA\)-homotopy in [\textit{X. Su}, ``Research on connected preserving mapping related to \(K\)-product topology'', Hebei Normal Univ. (2019)] and the present paper (or \(A\)-homotopy in [Han, Zbl 1377.68280]), use of it can be very limited in applied mathematics. Besides, the authors of the present paper are motivated by the structure of an \(H\)-topological space [\textit{S.-E. Han}, Filomat 31, No. 20, 6313--6328 (2017; Zbl 1499.54216)] and a ``Khalimsky space with a \(K\)-adjacency'' (\(KA\)-space, for brevity) in [Han, Zbl 1377.68280; \textit{S.-E. Han} and \textit{A. Šostak}, Comput. Appl. Math. 32, No. 3, 521--536 (2013; Zbl 1300.54012)]. However, the current paper has many things to be corrected and some parts were misunderstood by the authors, as follows: (1) First of all, the present paper contains several incorrect citations concerning the category of \(KAC\), the notions of a \(KA\)-homotopy and \(SC_{KA}\), the \(KA\)-contractibility of \(SC_{KA}^{n,4}\), and others, cf. [\textit{S.-E. Han}, Fixed Point Theory Appl. 2016, Paper No. 75, 20 p. (2016; Zbl 1350.55007); Han, Zbl 1377.68280; Han and Šostak, Zbl 1300.54012]. Furthermore, it misunderstood the notion of an \(H\)-topological space in [Han, Zbl 1499.54216], as follows: (1-1) As for \(SC_{KA}^{n,4}\) in Remark 3.1 of the present paper, the notion of simple closed \(KA\)-curve, or \(SC_{KA}^{n,l}\), should be attributed to [Han, Zbl 1377.68280; Han and Šostak, Zbl 1300.54012]. One can see the fact from the bottom of the page 532 of [Han and Šostak, Zbl 1300.54012] as well a…
reviewed by: Sang-Eon Han
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 68U10
Mathematics Subject Classification ID: 54H30
zbMATH DE Number: 7387386
zbMATH Keywords: digital space
zbMATH Keywords: \(SA\)-homotopy
zbMATH Keywords: \(KAC\)
zbMATH Keywords: \(KA\)-homotopy
zbMATH Keywords: \(H\)-topology
zbMATH Keywords: \(KA\)-contractibility
zbMATH Keywords: \(A\)-contractibility
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107739
OpenAlex ID: W3171355956
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DOI: 10.1016/J.TOPOL.2021.107739
description: scientific article; zbMATH DE number 7387386