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- W2955446890 abstract "For $X (subset R^n)$, assume the subspace $(X, E_X^n)$ induced by the $n$-dimensional Euclidean topological space $(R^n, E^n)$. Let $Z$ be the set of integers. Khalimsky topology on $Z$, denoted by $(Z, kappa)$, is generated by the set ${{2m-1, 2m, 2m+1},vert, m in {Z}}$ as a subbase. Besides, Khalimsky topology on $Z^n, n in N$, denoted by $(Z^n, kappa^n)$, is a product topology induced by $({Z}, kappa)$. Proceeding with a digitization of $(X, E_X^n)$ in terms of the Khalimsky ($K$-, for short) topology, we obtain a $K$-digitized space in ${Z}^n$, denoted by $D_K(X) (subset {Z}^n$), which is a $K$-topological space. Considering further $D_K(X)$ with $K$-adjacency, we obtain a topological graph related to the $K$-topology (a $KA$-space for short) denoted by $D_{KA}(X)$ (see an algorithm in Section 3). Motivated by an $A$-homotopy between $A$-maps for $KA$-spaces, the present paper establishes a new homotopy, called an $LA$-homotopy, which is suitable for studying homotopic properties of both $(X, E_X^n)$ and $D_{KA}(X)$ because a homotopy for Euclidean topological spaces has some limitations of digitizing $(X, E_X^n)$. The goal of the paper is to study some relationships among an ordinary homotopy equivalence for spaces $(X, E_X^n)$, an $LA$-homotopy equivalence for spaces $(X, E_X^n)$, and an $A$-homotopy equivalence for $KA$-spaces $D_{KA}(X)$. Finally, we classify $KA$-spaces (resp. $(X, E_X^n))$ via an $A$-homotopy equivalence (resp. an $LA$-homotopy equivalence). This approach can facilitate studies of applied topology, approximation theory and digital geometry." @default.
- W2955446890 created "2019-07-12" @default.
- W2955446890 creator A5028111620 @default.
- W2955446890 date "2020-02-06" @default.
- W2955446890 modified "2023-09-27" @default.
- W2955446890 title "Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces" @default.
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- W2955446890 doi "https://doi.org/10.15672/hujms.546983" @default.
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