Adjacency relations induced by some Alexandroff topologies on $ {\mathbb Z}^n $

Sang-Eon Han; Sang-Eon Han

doi:10.3934/math.2022645

AIMS Mathematics

2022, Volume 7, Issue 7: 11581-11596. doi: 10.3934/math.2022645

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Research article

Adjacency relations induced by some Alexandroff topologies on $ {\mathbb Z}^n $

Sang-Eon Han ^,

Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, Republic of Korea

Received: 26 January 2022 Revised: 28 March 2022 Accepted: 28 March 2022 Published: 14 April 2022
MSC : 54A05, 54J05, 68U05

Let $ (X, T) $ be an Alexandroff space. We define the adjacency relation $ AR_T $ on $ X $ induced by $ T $ as the irreflexive relation defined for $ x \neq y $ in $ X $ by:

$ (x,y) \in AR_T\,\,{\rm{if \;and\; only\; if}}\,\, x \in SN_T(y)\,\,{\rm{or}}\,\, y \in SN_T(x), $

where $ SN_T(z) $ is the smallest open set containing $ z $ in $ (X, T) $ and $ z \in \{x, y\} $. Two families of Alexandroff topologies $ (T_k, k \in {\mathbb Z}) $ and $ (T_k^\prime, k \in {\mathbb Z}) $ have been recently introduced on $ {\mathbb Z} $. The aim of this paper is to show that for each nonzero integers $ k $, the topologies $ T_k, T_k^\prime $, $ T_{-k} $, and $ T_{-k}^\prime $ are homeomorphic. The adjacency relations induced by the product topologies $ (T_k)^n $ and $ (T_k^\prime)^n $ are studied and compared with classical ones. We also show that the adjacency relations induced by $ T_k, T_k^\prime $, $ T_{-k} $, and $ T_{-k}^\prime $ are isomorphic. Then, note that the adjacency relations on $ {\mathbb Z} $ induced by these topologies, $ k \neq 0 $, are different from each other.
- adjacency relation,
- digital space,
- Khalimsky topology,
- Alexandroff topology,
- digital connectivity,
- $ T_{\frac{1}{2}} $-space,
- digital topology
Citation: Sang-Eon Han. Adjacency relations induced by some Alexandroff topologies on $ {\mathbb Z}^n $[J]. AIMS Mathematics, 2022, 7(7): 11581-11596. doi: 10.3934/math.2022645

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Abstract

Let $ (X, T) $ be an Alexandroff space. We define the adjacency relation $ AR_T $ on $ X $ induced by $ T $ as the irreflexive relation defined for $ x \neq y $ in $ X $ by:

$ (x,y) \in AR_T\,\,{\rm{if \;and\; only\; if}}\,\, x \in SN_T(y)\,\,{\rm{or}}\,\, y \in SN_T(x), $

where $ SN_T(z) $ is the smallest open set containing $ z $ in $ (X, T) $ and $ z \in \{x, y\} $. Two families of Alexandroff topologies $ (T_k, k \in {\mathbb Z}) $ and $ (T_k^\prime, k \in {\mathbb Z}) $ have been recently introduced on $ {\mathbb Z} $. The aim of this paper is to show that for each nonzero integers $ k $, the topologies $ T_k, T_k^\prime $, $ T_{-k} $, and $ T_{-k}^\prime $ are homeomorphic. The adjacency relations induced by the product topologies $ (T_k)^n $ and $ (T_k^\prime)^n $ are studied and compared with classical ones. We also show that the adjacency relations induced by $ T_k, T_k^\prime $, $ T_{-k} $, and $ T_{-k}^\prime $ are isomorphic. Then, note that the adjacency relations on $ {\mathbb Z} $ induced by these topologies, $ k \neq 0 $, are different from each other.

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