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

Abstract

Abstract. Let Z n be the Cartesian product of the set of integers Z and let (Z, T ) and (Z n , T n ) be the Khalimsky line topology on Z and the Khalimsky product topology on Z n , respectively. Then for a set X ⊂ Z n , consider the subspace (X, T n X ) induced from (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-kdeformation retract and investigate a classification of (computer topological) spaces X n,k in terms of a KD-(k 0 , k 1 )-homotopy equivalence.

Category Which Is Suitable for Studying Khalimsky Topological Spaces With Digital Connectivity

“…Indeed, this k(m, n)-adjacency is a generalization of the k-adjacency of [16]. Consequently, this operator leads to the k-adjacency relations of Z n [8] (for more details, see [9]):…”

“…Even though Khalimsky topology of Z n has strong merits of studying objects in Z n , a Khalimsky continuous map need not preserve digital connectivity [9]. However, if a map f : (X, T n 0 X ) → (Y, T n 1 Y ) performs Khalimsky continuity and digital connectivity, then it can be very useful in digital topology.…”

“…Let Z, N and Z n represent the sets of integers, natural numbers and points in the Euclidean n-dimensional space with integer coordinates, respectively. Motivated by the Alexandroff space [1], the Khalimsky nD space (briefly, (Z n , T n )) was established and the study of its properties includes the papers [9,12,15]. In this paper we consider a subset X ⊂ Z n as a subspace of (Z n , T n ), denoted by (X, T n X ), n ≥ 1 [5,9,10].…”

“…In relation to the study of subsets of the Euclidean nD space with integer coordinates, we have often used the Khalilmsky topological structure [5,9,10,12,15]. Let Z, N and Z n represent the sets of integers, natural numbers and points in the Euclidean n-dimensional space with integer coordinates, respectively.…”

“…However, if a map f : (X, T n 0 X ) → (Y, T n 1 Y ) performs Khalimsky continuity and digital connectivity, then it can be very useful in digital topology. Thus, we have often studied a Khalimsky topological space with digital k-connectivity of Z n , n ∈ N [9,10] (see Definition 1) and further, used a map preserving digital connectivity [5].…”