Homotopic properties ofKA-digitizations ofn-dimensional Euclidean spaces

Abstract

ForX(⊂$R^n$), assume the subspace(X, $E_n{^X})induced by then-dimensional Euclideantopological space(Rn, En). LetZbe the set of integers. Khalimsky topology onZ,denoted by(Z, κ), is generated by the set{{2m−1,2m,2m+ 1}|m∈Z}as a subbase.Besides, Khalimsky topology on$Z^n$, n∈N, denoted by($Z^n, κ^n$), is a product topologyinduced by(Z, κ). Proceeding with a digitization of ($X, E^n_{X}$)in terms of the Khalimsky(K-, for short) topology, we obtain aK-digitized space in$Z^n$, denoted byDK(X)(⊂$Z^n$),which is aK-topological space. Considering further$D_K$(X)withK-adjacency, we obtain atopological graph related to theK-topology(aKA-space for short) denoted by$D_{KA}$(X)(seean algorithm in Section 3). Motivated by anA-homotopy betweenA-maps forKA-spaces,the present paper establishes a new homotopy, called anLA-homotopy, which is suitablefor studying homotopic properties of both(X, EnX)andDKA(X)because a homotopyfor Euclidean topological spaces has some limitations of digitizing($X, E^n_{X}$). The goal ofthe paper is to study some relationships among an ordinary homotopy equivalence forspaces($X, E^n_{X}$), anLA-homotopy equivalence for spaces(X, EnX), and anA-homotopyequivalence forKA-spaces$D_{KA}$(X). Finally, we classifyKA-spaces (resp.($X, E^n_{X}$))viaanA-homotopy equivalence (resp.anLA-homotopy equivalence). This approach canfacilitate studies of applied topology, approximation theory and digital geometry.