KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS

Let Zⁿ be the Cartesian product of the set of integers Z and let ( Z , T) and ( Zⁿ , Tⁿ ) be the Khalimsky line topology on Z and the Khalimsky product topology on Zⁿ , respectively. Then for a set X\;⊂\;Zⁿ , consider the subspace (X, TⁿX ) induced from ( Zⁿ , Tⁿ ). Considering a k-adjacency on (X, TⁿX ), we call it a (computer topological) space with k-adjacency and use the notation (X, k, TⁿX ) := Xn,k . In this paper we introduce the notions of KD-( k₀ , k₁ )-homotopy equivalence and KD-k-deformation retract and investigate a classification of (computer topological) spaces Xn,k in terms of a KD-( k₀ , k₁ )-homotopy equivalence.