Reversibility of Tk-topological spaces and the universality problem for the collection of Tk-topological spaces

As a generalization of the Khalimsky line topological space, consider the topological space (Z, Tk) (resp. (Z, T?k)) on the set of integers, where the topology Tk (resp. T?k) is generated by the set Sk (resp. S?k) as a subbase, k ? Z, and Sk:= {Sk,t | Sk,t:= {2t, 2t + 1, 2t + 2k + 1}, t ? Z} (resp. S?k:= {S?k,t | S?k,t:= {2t,2t+1,2t + 2k } ,t? Z }).For k ? Z \ {0}, each of the Tk- and T? k-topological space indeed satisfies the T1/2-separation axiom. Besides, for k ? Z \ {0}, each of (Z, Tk) and (Z, T?k) is an Alexandroff space and is neither a Kuratowski space nor a regular space. The paper initially proves that each of (Z, Tk) and (Z, T?k), k ? Z, is reversible. Next, let T be the collection of Tk-topological spaces {(Z, Tk), k ? Z \ {0}}, and T? be the set of T?k-topological spaces {(Z, T?k), k ? Z \ {0}}. Then the paper deals with an existence problem of a universal element in T and T?.