An extension problem of a connectedness preserving map between Khalimsky spaces

Abstract

The goal of the present paper is to study the extension problem of a connected preserving (for short, {\it CP}-) map between Khalimsky spaces.  As a generalization of a Khalimsky continuous map, for Khalimsky spaces the recent paper \cite{HS1} develops a function sending connected sets to connected ones (for brevity, an $A$-map: see Definition 3 in the present paper). Since this map  can play an important role in applied topology including digital topology, digital geometry and mathematical morphology, the present paper studies the extension problem of a {\it CP}-map in terms of both an $A$-retract and an $A$-isomorphism (see Example 5.2 as a motivating example).  Since Khalimsky topological spaces have been often used for studying digital images, this extension problem can contribute to computer science areas and mathematical morphology.