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

Abstract

For X(⊂ R n ), assume the subspace (X, E n X ) 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, κ), is generated by the set {{2m − 1, 2m, 2m + 1} | m ∈ Z} as a subbase. Besides, Khalimsky topology on Z n , n ∈ N, denoted by (Z n , κ n ), is a product topology induced by (Z, κ). Proceeding with a digitization of (X, E n X ) in terms of the Khalimsky (K-, for short) topology, we obtain a K-digitized space in Z n , denoted by D K (X)(⊂ 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 n X ) and D KA (X) because a homotopy for Euclidean topological spaces has some limitations of digitizing (X, E n X ). The goal of the paper is to study some relationships among an ordinary homotopy equivalence for spaces (X, E n X ), an LA-homotopy equivalence for spaces (X, E n X ), and an A-homotopy equivalence for KA-spaces D KA (X). Finally, we classify KA-spaces (resp. (X, E n X )) via an A-homotopy equivalence (resp. an LA-homotopy equivalence). This approach can facilitate studies of applied topology, approximation theory and digital geometry.