36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

"Many types of digital connectivities derived from some Alexandroff topologies on ${\mathbb Z}^n$"

Han, Sang-Eon

Recently, 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.

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