Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

HOME ALL ARTICLES View

J. Korean Math. Soc. 2010; 47(5): 1031-1054

Printed September 1, 2010

https://doi.org/10.4134/JKMS.2010.47.5.1031

Copyright © The Korean Mathematical Society.

KD-$(k_0, k_1)$-homotopy equivalence and its applications

Sang-Eon Han

Chonbuk National University

Abstract

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 $(X, T_X^n)$ induced from $({\mathbb Z}^n, T^n)$. Considering a $k$-adjacency on $(X, T_X ^n)$, we call it a (computer topological) space with $k$-adjacency and use the notation $(X, k, T_X ^n):=X_{n, k}$. 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.

Keywords: computer topology, digital topology, digital space, KD-$(k_0, k_1)$-continuity, KD-$k$-deformation retract, digital homotopy equivalence, KD-$(k_0, k_1)$-homotopy equivalence, KD-$k$-homotopic thinning

MSC numbers: 55P10, 55P15, 55Q70, 54A10, 65D18, 68U05, 68U10