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

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 $x\ne y$ in $X$ by:

$$(x,y)\in A{R}_T\mathrm{i}\mathrm{f}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{f}x\in S{N}_T(y)\mathrm{o}\mathrm{r}y\in S{N}_T(x),$$

where $S{N}_T(z)$ 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 $(T_k^{\mathrm{\prime }},k\in \mathbb{Z})$ have been recently introduced on $\mathbb{Z}$. The aim of this paper is to show that for each nonzero integers $k$, the topologies $T_k,T_k^{\mathrm{\prime }}$, $T_{-k}$, and $T_{-k}^{\mathrm{\prime }}$ are homeomorphic. The adjacency relations induced by the product topologies $(T_k{)}^n$ and $(T_k^{\mathrm{\prime }}{)}^n$ are studied and compared with classical ones. We also show that the adjacency relations induced by $T_k,T_k^{\mathrm{\prime }}$, $T_{-k}$, and $T_{-k}^{\mathrm{\prime }}$ are isomorphic. Then, note that the adjacency relations on $\mathbb{Z}$ induced by these topologies, $k\ne 0$, are different from each other.