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

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: \begin{document}$ (x,y) \in AR_T\,\,{\rm{if \;and\; only\; if}}\,\, x \in SN_T(y)\,\,{\rm{or}}\,\, y \in SN_T(x), $\end{document} 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.