• 호남수학학술지
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• 제37권4호
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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.

• 대한수학회지
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• 제45권4호
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• pp.923-952
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• 2008

In this paper several continuities and homeomorphisms in computer topology are studied and their applications are investigated in relation to the classification of subs paces of Khalimsky n-dimensional space $({\mathbb{Z}}^n,\;T^n)$. Precisely, the notions of K-$(k_0,\;k_1)$-,$(k_0,\;k_1)$-,KD-$(k_0,\;k_1)$-continuities, and Khalimsky continuity as well as those of K-$(k_0,\;k_1)$-, $(k_0,\;k_1)$-, KD-$(k_0,\;k_1)$-homeomorphisms, and Khalimsky homeomorphism are studied and further, their applications are investigated.

• 호남수학학술지
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• 제28권4호
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• pp.591-603
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• 2006

For a set $X{\subset}{\mathbb{Z}}^n$ let $(X,\;T^n_X)$ be the subspace of the Khalimsky n-dimensional space $({\mathbb{Z}}^n,\;T^n)$, $n{\in}N$. Considering a k-adjacency of $(X,\;T^n_X)$, we use the notation $(X,\;k,\;T^n_X)$. In this paper for a map $$f:(X,\;k,\;T^n_X){\rightarrow}(Y,\;2\;T_Y)$$, we find the condition that weak (k, 2)-continuity of the map f implies strong (k, 2)-continuity of f.

• 대한수학회보
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• 제47권5호
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• pp.915-932
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• 2010

The goal of this paper is to study extension problems of several continuities in computer topology. To be specific, for a set $X\;{\subset}\;Z^n$ take a subspace (X, $T_n^X$) induced from the Khalimsky nD space ($Z^n$, $T^n$). Considering (X, $T_n^X$) with one of the k-adjacency relations of $Z^n$, 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.

• 호남수학학술지
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• 제33권2호
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• pp.231-246
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• 2011

Let $X_{n,k}$ be a Khalimsky topological n dimensional subspace with digital k-connectivity. In relation to the classification of spaces $X_{n,k}$, by comparing several kinds of continuities and homeomorphisms, the paper proposes a category which is suitable for studying the spaces $X_{n,k}$.

• 호남수학학술지
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• 제34권3호
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• pp.451-465
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• 2012

Since there are several kinds of continuities of maps between digital spaces, the paper compares them, which can play an important role in digital topology and discrete geometry.

• 대한수학회지
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• 제47권5호
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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.

• 호남수학학술지
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• 제38권3호
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• pp.625-642
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• 2016

The recent papers [7, 12] developed two maps named by an LA- [7] and an LMA-map [12]. The present paper studies their properties and further, develops generalized versions of an LA- and an LMA-map, which makes the LA- [7] and the LMA-map [12] improved.

• 호남수학학술지
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• 제33권4호
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• pp.617-628
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• 2011

In relation to the classification of finite topological spaces the paper [17] studied various properties of finite topological spaces. Indeed, the study of future internet system can be very related to that of locally finite topological spaces with some order structures such as preorder, partial order, pretopology, Alexandroff topological structure and so forth. The paper generalizes the results from [17] so that the paper can enlarge topological and homotopic properties suggested in the category of finite topological spaces into those in the category of locally finite topological spaces including ALF spaces.

• 호남수학학술지
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• 제35권4호
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• pp.707-716
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• 2013

The present paper consists of two parts. Since the recent paper [4] proved that an Alexandroff $T_0$-space is a semi-$T_{\frac{1}{2}}$-space, the first part studies semi-open and semi-closed structures of the Khalimsky nD space. The second one focuses on the study of a relation between the LS-property of ($SC^{n_1,l_1}_{k_1}{\times}SC^{n_2,l_2}_{k_2}$, k) relative to the simple closed $k_i$-curves $SC^{n_i,l_i}_{k_i}$, $i{\in}\{1,2\}$ and its normal k-adjacency. In addition, the present paper points out that the main theorems of Boxer and Karaca's paper [3] such as Theorems 4.4 and 4.7 of [3] cannot be new assertions. Indeed, instead they should be attributed to Theorems 4.3 and 4.5, and Example 4.6 of [10].