Strong homotopy in finite topological adjacency category
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 as Sections 4 of [Han, Zbl 1377.68280]. (1-2) Definition 3.1 of this paper should be attributed to [Han, Zbl 1377.68280] instead of the work of [Su, loc. cit.]. One can see that Su [loc. cit.] used a \(KA\)-homotopy and a category \(KAC\), the \(KA\)-contractibility of \(SC_{KA}^{n,4}\) and others in her/his work without any citation. See the \(KA\)-homotopy from Definition 5.1 of [Han, Zbl 1377.68280] and the category \(KAC\) in [Han, Zbl 1377.68280; Han and Šostak, Zbl 1300.54012]. (1-3) As for the \(K\)-homotopy in [Han, Zbl 1350.55007] (see the first line on the page 6 of the present paper or Definition 6 of [Han, Zbl 1350.55007]), the authors of the present paper misunderstood the notion. As for the \(K\)-homotopy, the paper [Han, Zbl 1350.55007] wrote \([a,b]_{\mathbb{Z}} \in \{[0, m]_{\mathbb{Z}}, [1, m+1]_{\mathbb{Z}}\}\) and the set \(X \times [a,b]_{\mathbb{Z}}\) does not require any product topological structure. Namely, the set \(X \times [a,b]_{\mathbb{Z}}\) is just a Cartesian product from the given two sets. This means that the number \(a\) of the digital interval \([a,b]_{\mathbb{Z}}\) can be considered as an even or odd number. One can easily see that if we suppose \(X \times [a,b]_{\mathbb{Z}}\) as a product space, the space \(X \times [1, m+1]_{\mathbb{Z}}\) cannot be a Khalimsky topological space in general. Thus the above notation of [Han, Zbl 1350.55007] is correct. Thus, Theorem 2.3 of the present paper is not related to the \(K\)-homotopy of [Han, Zbl 1350.55007] at all. Namely, this assertion is another story that is not related to the \(K\)-homotopy. (1-4) As for the notion of a \(KA\)-homotopy, it should be attributed to the paper [Han, Zbl 1377.68280] instead of [Su, loc. cit.] and the present paper. The notion was already established and published in 2015--2016 with Definition 5.1 (pages 1272--1273 of [Han, Zbl 1377.68280]). (1-5) The present paper mentioned the \(H\)-topology in [\textit{Z. Zhang} et al., Topology Appl. 300, Article ID 107739, 13 p. (2021; Zbl 07387386)] (see the lines from 15 to 20 at the page 3 of the present paper). More precisely, the present paper referred to several examples for Alexandroff topological spaces such as finite topological spaces [\textit{J. P. May}, Finite topological spaces. Notes for REU (2003; \url{http://www.math.uchicago.edu/~may/MISCMaster.html})], generalized \(MW\)-topological spaces on \({\mathbb{Z}}^n\) (or \(H\)-topological spaces) [Han, Zbl 1499.54216], and Khalimsky topological spaces [\textit{E. Khalimsky}, J. Appl. Math. Simulation 1, No. 1, 25--40 (1987; Zbl 0655.68021)]. However, the present paper seriously misunderstood the notion of an \(H\)-topological space [Han, Zbl 1499.54216]. The present paper wrote that an \(H\)-topological space \((X, \gamma_X^n)\) in [Han, Zbl 1499.54216] is generated by the set \(\mathcal{B}=\{SN_X(p)\mid p\in X\}\) as a base (see page 3 of the present paper[), where for each point \(p:=(x_1, x_2, \dots, x_n) \in {\mathbb{Z}}^n\). Indeed, the set \(AN_{2n}(p)\) of the present paper in \((X, \gamma_X^n)\) is equal to \(N_{2n}(p,1) \subset X \subset {\mathbb{Z}}^n\) in [\textit{S.-E. Han}, Topology Appl. 196, Part B, 468--482 (2015; Zbl 1353.54004); Han, Zbl 1499.54216]. To be specific, in \((X, \gamma_X^n)\) we obtain the following identity (see [Han, Zbl 1499.54216]): \[ AN_{2n}(p)=\{x \in X\mid x\text{ is }2n\text{-adjacenct to } p\} \cup \{p\}. \] Based on this observation, we now correct some mistakes occurred in the present paper, as follows: Let us consider the topological space \((X, T_{\mathcal{B}})\) generated by the set \(\mathcal{B}=\{SN_X(p)\mid p\in X\}\) as a base, denoted by \((X, T_{\mathcal{B}})\) as usual, \(X \subset {\mathbb{Z}}^n\). Then, for \(X\subset {\mathbb{Z}}^n\), this topology \(T_{\mathcal{B}}\) is clearly different from the \(H\)-topology \(\gamma_X^n\) in [Han, Zbl 1499.54216]. Thus the topological space \((X, T_{\mathcal{B}})\) is not an \(H\)-topological \((X, \gamma_X^n)\), which invokes some errors.
scientific article; zbMATH DE number 7387386
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