### An extension problem of a connectedness preserving map between Khalimsky spaces

Sang-Eon Han
2016 Filomat
The goal of the present paper is to study an extension problem of a connected preserving (for short, CP-) map between Khalimsky (K-for brevity, if there is no ambiguity) spaces. As a generalization of a K-continuous map, for K-topological spaces the recent paper [13] develops a function sending connected sets to connected ones (for brevity, an A-map: see Definition 3.1 in the present paper). Since this map plays an important role in applied topology including digital topology, digital geometry
more » ... , digital geometry and mathematical morphology, the present paper studies an extension problem of a CP-map in terms of both an A-retract and an A-isomorphism (see Example 5.2). Since K-topological spaces have been often used for studying digital images, this extension problem can contribute to a certain areas of computer science and mathematical morphology. (Sang-Eon Han) By using the K-adjacency, we define the following terminology: Definition 2.3. [13] For a space (X, T n X ) := X we define the following: (1) Two distinct points x, y ∈ X are called KA-connected if there is an injective sequence (or path) (x i ) i∈[0,m] Z on X with {x 0 = x, x 1 , ..., x m = y} such that x i and x i+1 are K-adjacent, i ∈ [0, m − 1] Z , m ≥ 1. This sequence is called a KA-path. Furthermore, the number m is called the length of this KA-path. Furthermore, a KA-path is called a closed KA-curve if x 0 = x m . (2) A simple KA-path on X is a KA-path such that x i and x j are K-adjacent if and only if | i − j | = 1. Furthermore, we say that a simple closed KA-curve with m elements (x i ) i∈[0,m] Z is a simple KA-path with x 0 = x m such that x i and x j are K-adjacent if and only if either j = i + 1(mod m) or i = j + 1(mod m), m ≥ 4.