$U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces

Abstract

For $X\subset R^n$ let $(X, E_X^n)$ be the usual topological space induced by the $n$D Euclidean topological space $(R^n, E^n)$. Based on the upper limit ($U$-, for short) topology (resp. the lower limit ($L$-, for brevity) topology), after proceeding with a digitization of $(X, E_X^n)$, we obtain a $U$- (resp. an $L$-) digitized space denoted by $D_U(X)$ (resp. $D_L(X)$) in $Z^n$ [16]. Further considering $D_U(X)$ (resp. $D_L(X)$) with a digital $k$-connectivity, we obtain a digital image from the viewpoint of digital topology in a graph-theoretical approach, i.e. Rosenfeld model [25], denoted by $D_{U(k)}(X)$ (resp. $D_{L(k)}(X)$) in the present paper. Since a Euclidean topological homotopy has some limitations of studying a digitization of $(X, E_X^n)$, the present paper establishes the so called $U(k)$-homotopy (resp. $L(k)$-homotopy) which can be used to study homotopic properties of both $(X, E_X^n)$ and $D_{U(k)}(X)$ (resp. both $(X, E_X^n)$ and $D_{L(k)}(X)$). The goal of the paper is to study some relationships among an ordinary homotopy equivalence, a $U(k)$-homotopy equivalence, an $L(k)$-homotopy equivalence and $k$-homotopy equivalence. Finally, we classify $(X, E_X^n)$ in terms of a $U(k)$-homotopy equivalence and an $L(k)$-homotopy equivalence. This approach can be used to study applied topology, approximation theory and digital geometry. 

Keywords

• $U(k)$-digitization,$L(k)$-digitization,$U$- and $L$-localized neighborhood,$U(k)$- and $L(k)$- homotopy

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