Reversibility of $T_k$-topological spaces and the universality problem for the collection of $T_k$-topological spaces

Abstract

As a generalization of the Khalimsky line topological space, consider the topological space $({\mathbb Z}, T_k)$ (resp. $({\mathbb Z}, T_k^\prime)$) on the set of integers, where the topology $T_k$(resp. $T_k^\prime$) is generated by the set $S_k$ (resp. $S_k^\prime$) as a subbase, $k \in {\mathbb Z}$, and $S_k:=\{S_{k, t}\,\vert \, S_{k, t}:=\{2t, 2t+1, 2t+2k+1\},\, t \in {\mathbb Z}\}$ (resp. $S_k^\prime:=\{S_{k, t}^\prime\,\vert \, S_{k, t}^\prime:=\{2t, 2t+1, 2t+2k\},\, t \in {\mathbb Z}\}$).
For $k \in {\mathbb Z}\setminus \{0\}$, each of the $T_k$- and $T_k^\prime$-topological space indeed satisfies the $T_{\frac{1}{2}}$-separation axiom.
Besides, for $k \in {\mathbb Z}$, each of $({\mathbb Z}, T_k)$ and $({\mathbb Z}, T_k^\prime)$ is an Alexandroff space and is neither a Kuratowski space nor a regular space. The paper initially proves that each of $({\mathbb Z}, T_k)$ and $({\mathbb Z}, T_k^\prime), k \in {\mathbb Z}$, is reversible.
Next, let $\mathcal{T}$ be the collection of $T_k$-topological
spaces $\{({\mathbb Z}, T_k), k \in {\mathbb Z} \setminus \{0\}\}$, and $\mathcal{T}^\prime$ be the set of $T_k^\prime$-topological spaces $\{({\mathbb Z}, T_k^\prime), k \in {\mathbb Z} \setminus \{0\}\}$. Then the paper deals with an existence problem of a universal element in $\mathcal{T}$ and $\mathcal{T}^\prime$.