Bulletin of the
Korean Mathematical Society BKMS

The goal of this paper is to study extension problems of several continuities in computer topology.To be specific, for a set $X \subset {\bf Z}^n$ take a subspace $(X, T_X^n)$ induced from the Khalimsky $n$D space $({\bf Z}^n, T^n)$. Considering $(X, T_X ^n)$ with one of the $k$-adjacency relations of ${\bf Z}^n$, we call it a computer topological space (or a space if not confused) denoted by $X_{n, k}$. In addition, we introduce several kinds of $k$-retracts of $X_{n, k}$, investigate their properties related to several continuities and homeomorphisms in computer topology and study extension problems of these continuities in relation with these $k$-retracts.

Keywords: computer topology, digital topology, extension problem, Khalimsky topology, computer topological continuity, computer topological homeomorphism, $k$-retract

MSC numbers: 54C20, 54C05, 54D05, 54F05, 68U05