Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces

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.