U(k)- and L(k)-homotopic properties of digitizations of nD Hausdorff spaces

For X ⊂ Rn let (X, En X) be the usual topological space induced by the nD Euclidean topological space (Rn , En ). Based on the upper limit (U-, for short) topology (resp. the lower limit (L-, for brevity) topology), after proceeding with a digitization of (X, En X), we obtain a U- (resp. an L-) digitized space denoted by DU (X) (resp. DL(X)) in Z n [16]. Further considering DU (X) (resp. DL(X)) with a digital k-connectivity, we obtain a digital image from the viewpoint of digital topology in a graph-theoretical approach, i.e. Rosenfeld model [25], denoted by DU(k)(X) (resp. DL(k)(X)) in the present paper. Since a Euclidean topological homotopy has some limitations of studying a digitization of (X, En X), the present paper establishes the so called U(k)-homotopy (resp. L(k)-homotopy) which can be used to study homotopic properties of both (X, En X) and DU(k)(X) (resp. both (X, En X) and DL(k)(X)). The goal of the paper is to study some relationships among an ordinary homotopy equivalence, a U(k)-homotopy equivalence, an L(k)-homotopy equivalence and a k-homotopy equivalence. Finally, we classify (X, En X) in terms of a U(k)-homotopy equivalence and an L(k)-homotopy equivalence. This approach can be used to study applied topology, approximation theory and digital geometry.

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