Compression of Khalimsky topological spaces

Abstract

Aiming at the study of the compression of Khalimsky topological spaces which is an interesting field in digital geometry and computer science, the present paper develops a new homotopy thinning suitable for the work. Since Khalimsky continuity of maps between Khalimsky topological spaces has some limitations of performing a discrete geometric transformation, the paper uses another continuity (see Definition 3.4) that can support the discrete geometric transformation and a homotopic thinning suitable for studying Khalimsky topological spaces. By using this homotopy, we can develop a new homotopic thinning for compressing the spaces and can write an algorithm for compressing 2D Khalimsky topological spaces.

Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

“…As an Alexandroff space, the Khalimsky nD space was established and the study of its properties includes the papers [8,[18][19][20][21]. Let us now recall basic notions from the K-topology on Z n .…”

The Fixed Point Property of Non-Retractable Topological Spaces

“…Hereafter, for a subset X ⊂ Z n we will denote by (X, κ n X ), n ≥ 1, a subspace induced by (Z n , κ n ) and it is called a K-topological space. The study of these spaces includes References [4,28,[32][33][34][35][36][37][38][39].…”

The fixed point property of the smallest open neighborhood of the n-dimensional Khalimsky topological space

“…again if there is no danger of the ambiguity. In (Z n , κ n ), let us now recall some properties of the K-continuity of maps between two K-topological spaces [14][15][16] as follows: for two K-topological spaces (X, κ n 0 X ) := X and (Y, κ n 1 Y ) := Y, a function f : X → Y is said to be K-continuous at a point x ∈ X if f is continuous at the point x from the viewpoint of Khalimsky product topology as usual, i.e. f (SN…”