**"Many types of digital connectivities derived from some Alexandroff topologies on ${\mathbb Z}^n$"**
#### Han, Sang-EonRecently, countably many kinds of Alexandroff topologies $(T_k, k \in {\mathbb Z})$ and $(T_k^\prime, k \in {\mathbb Z})$ have been established on ${\mathbb Z}$.
In this talk, we will 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 investigated and compared with the adjacency derived from the $n$-dimensional Khalimsky topological space. Besides, 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, which makes the earlier works more advanced.
Based on this approach, we will show some strong advantages of taking a suitable adjacency of
${\mathbb Z}^n$ according to our needs.
Thus we can apply this approach into digital image processing, mathematical morphology, and so on. |