• Honam Mathematical Journal
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• v.37 no.4
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• pp.577-593
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• 2015

Several kinds of homotopies have been substantially used to study topological properties of digital spaces. The present paper, as a survey article, studies some recent results in the field of homotopy theory associated with Khalimsky topology. In particular, Khalimsky topological properties of digital products related to the establishment of the homotopies are mainly treated.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1031-1054
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• 2010

Let $\mathbb{Z}^n$ be the Cartesian product of the set of integers $\mathbb{Z}$ and let ($\mathbb{Z}$, T) and ($\mathbb{Z}^n$, $T^n$) be the Khalimsky line topology on $\mathbb{Z}$ and the Khalimsky product topology on $\mathbb{Z}^n$, respectively. Then for a set $X\;{\subset}\;\mathbb{Z}^n$, consider the subspace (X, $T^n_X$) induced from ($\mathbb{Z}^n$, $T^n$). Considering a k-adjacency on (X, $T^n_X$), we call it a (computer topological) space with k-adjacency and use the notation (X, k, $T^n_X$) := $X_{n,k}$. In this paper we introduce the notions of KD-($k_0$, $k_1$)-homotopy equivalence and KD-k-deformation retract and investigate a classification of (computer topological) spaces $X_{n,k}$ in terms of a KD-($k_0$, $k_1$)-homotopy equivalence.