@article{Han2022, 
author = {Sang-Eon Han},
title = {Adjacency relations induced by some Alexandroff topologies on                      Z              n},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {7},
pages = {11581-11596},
keywords = {adjacency relation, digital space, Khalimsky topology, Alexandroff topology, digital connectivity, T1/2-space, digital topology},
url = {https://www.sciopen.com/article/10.3934/math.2022645},
doi = {10.3934/math.2022645},
abstract = {Let    (  X  ,  T  ) be an Alexandroff space. We define the adjacency relation    A      R    T   on    X induced by    T as the irreflexive relation defined for    x  ≠  y in    X by:     (  x  ,  y  )  ∈  A      R    T                  i      f            a      n      d            o      n      l      y            i      f            x  ∈  S      N    T    (  y  )                o      r            y  ∈  S      N    T    (  x  )  ,where    S      N    T    (  z  ) is the smallest open set containing    z in    (  X  ,  T  ) and    z  ∈  {  x  ,  y  }. Two families of Alexandroff topologies    (      T    k    ,  k  ∈            Z        ) and    (      T    k    ′    ,  k  ∈            Z        ) have been recently introduced on              Z      . The aim of this paper is to show that for each nonzero integers    k, the topologies        T    k    ,      T    k    ′  ,        T          −      k      , and        T          −      k        ′   are homeomorphic. The adjacency relations induced by the product topologies    (      T    k        )    n   and    (      T    k    ′        )    n   are studied and compared with classical ones. We also show that the adjacency relations induced by        T    k    ,      T    k    ′  ,        T          −      k      , and        T          −      k        ′   are isomorphic. Then, note that the adjacency relations on              Z       induced by these topologies,    k  ≠  0, are different from each other.}
}