KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS
Sang-Eon Han · Journal of the Korean Mathematical Society · 2010
Let $\mathbb{Z}^n$ be the Cartesian product of the set of integers $\mathbb{Z}$ and let ( $\mathbb{Z}$ , T) and ( $\mathbb{Z}^n$ , $T^n$ ) be the Khalimsky line topology on $\mathbb{Z}$ and the Khalimsky product topology on $\mathbb{Z}^n$ , respectively. Then for a set $X\;{\subset}\;\mathbb{Z}^n$ , consider the subspace (X, $T^n_X$ ) induced from ( $\mathbb{Z}^n$ , $T^n$ ). Considering a k-adjacency on (X, $T^n_X$ ), we call it a (computer topological) space with k-adjacency and use the notation (X, k, $T^n_X$ ) := $X_{n,k}$ . In this paper we introduce the notions of KD-( $k_0$ , $k_1$ )-homotopy equivalence and KD-k-deformation retract and investigate a classification of (computer topological) spaces $X_{n,k}$ in terms of a KD-( $k_0$ , $k_1$ )-homotopy equivalence.