Embeddings of the $T_k$-topological spaces

Abstract

Let $({\mathbb Z}, T_k)$ be a topological space on the set of integers, where the topology $T_k$ is generated by the set $S_k$ as a subbase, where $k \in {\mathbb Z}$ and $S_k:=\{S_{k, t}\,\vert \, S_{k, t}:=\{2t, 2t+1, 2t+2k+1\},\, t \in {\mathbb Z}\}$. Then we call $({\mathbb Z}, T_k)$ a $T_k$-topological space. For the set of positive integers $k_1, k_2 \in {\mathbb N}$, the paper initially proves that $({\mathbb Z}, T_{k_1})$ is topologically embedded into $({\mathbb Z}, T_{k_2})$ if and only if $k_1 \leq k_2$, i.e., $\exists$ an embedding $h_{(k_1,k_2)}: ({\mathbb Z}, T_{k_1}) \to ({\mathbb Z}, T_{k_2}) \Leftrightarrow k_1 \leq k_2$. Furthermore, in the case of $k_1 \leq k_2$, the subspace $({\mathbb Z} \setminus Im(h_{(k_1, k_2)}), (T_{k_2})_{{\mathbb Z} \setminus Im(h_{(k_1, k_2)})})$ induced from $({\mathbb Z}, T_{k_2})$ has $(k_2-k_1)$ components. Besides, each of the components is homeomorphic with the Khalimsky ($K$-, for brevity) topological line. Finally, we obtain some embeddings of $({\mathbb Z}^n, \kappa^n)$ into $({\mathbb Z}^n, (T_k)^n)$, the $n$-fold product topological space of a $T_k$-topological space, $k \in {\mathbb N}$, where $({\mathbb Z}^n, \kappa^n)$ is the $n$-dimensional $K$-topological space. Since a $T_k$-topological space is an Alexandroff space, an embedding problem of a $T_k$-topological space plays an important role in pure and applied topology.