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## Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces

#### Sang-eon HAN [1]

For $X (\subset R^n)$, assume the subspace $(X, E_X^n)$ induced by the $n$-dimensional Euclidean topological space $(R^n, E^n)$. Let $Z$ be the set of integers. Khalimsky topology on $Z$, denoted by  $(Z, \kappa)$, is generated by the set $\{\{2m-1, 2m, 2m+1\}\,\vert\, m \in {Z}\}$ as a subbase. Besides, Khalimsky topology on  $Z^n, n \in N$, denoted by $(Z^n, \kappa^n)$, is a product topology induced by $({Z}, \kappa)$. Proceeding with a digitization of $(X, E_X^n)$ in terms of the Khalimsky ($K$-, for short) topology, we obtain a $K$-digitized space in ${Z}^n$, denoted by $D_K(X) (\subset {Z}^n$), which is a $K$-topological space. Considering further $D_K(X)$ with $K$-adjacency, we obtain a topological graph related to the $K$-topology (a $KA$-space for short) denoted by $D_{KA}(X)$ (see an algorithm in Section 3). Motivated by an $A$-homotopy between $A$-maps for $KA$-spaces,  the present paper establishes a new homotopy, called an $LA$-homotopy, which is suitable for studying homotopic properties of both $(X, E_X^n)$ and $D_{KA}(X)$ because a homotopy for Euclidean topological spaces has some limitations of digitizing $(X, E_X^n)$. The goal of the paper is to study some relationships among an ordinary homotopy equivalence for spaces $(X, E_X^n)$, an $LA$-homotopy equivalence for spaces $(X, E_X^n)$, and an $A$-homotopy equivalence for $KA$-spaces $D_{KA}(X)$. Finally, we classify  $KA$-spaces (resp. $(X, E_X^n))$ via an $A$-homotopy equivalence (resp. an $LA$-homotopy equivalence). This approach can facilitate studies of applied topology, approximation theory and digital geometry.
Digital topology, KA-digitization, Khalimsky adjacency, A-map, LA-map, K-topological graph, K-localized neighborhood, LA-homotopy equivalence, A-homotopy equivalence
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Primary Language en Mathematics Mathematics Orcid: 0000-0002-8030-8253Author: Sang-eon HAN Institution: Chonbuk National UniversityCountry: South Korea Publication Date : February 6, 2020
 Bibtex @research article { hujms546983, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {236 - 253}, doi = {10.15672/hujms.546983}, title = {Homotopic properties of \$KA\$-digitizations of \$n\$-dimensional Euclidean spaces}, key = {cite}, author = {HAN, Sang-eon} } APA HAN, S . (2020). Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces. Hacettepe Journal of Mathematics and Statistics , 49 (1) , 236-253 . DOI: 10.15672/hujms.546983 MLA HAN, S . "Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 236-253 Chicago HAN, S . "Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 236-253 RIS TY - JOUR T1 - Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces AU - Sang-eon HAN Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.546983 DO - 10.15672/hujms.546983 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 236 EP - 253 VL - 49 IS - 1 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.546983 UR - https://doi.org/10.15672/hujms.546983 Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces %A Sang-eon HAN %T Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 1 %R doi: 10.15672/hujms.546983 %U 10.15672/hujms.546983 ISNAD HAN, Sang-eon . "Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces". Hacettepe Journal of Mathematics and Statistics 49 / 1 (February 2020): 236-253 . https://doi.org/10.15672/hujms.546983 AMA HAN S . Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 236-253. Vancouver HAN S . Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 253-236.

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