I'm still relatively new to topology, so I haven't fully caught on to terminological conventions. In particular, I've have encountered phrasing about point-sets which is slightly unclear to me.
Given $S$ is the point set for $X$ and $Y$ where the topologies for $X$ and $Y$ are $\tau_1$ and $\tau_2$ respectively.
Am I interpreting it correctly to think $X = (S,\tau_1)$ and $Y = (S,\tau_2)$ where the underlying spaces are the same, i.e $S$?
A point set A, of a space (S,T) is a subset of S.
Point sets are distinguished from other sets such
as a collection of closed sets.
As a set, A does not have a topology.
A can be made into a topology by giving it the subspace
topology, $T_A$ = { U $\cap$ A : U in T }, which is not T.
As T is not a topology for A, (A,T) is wrongthink.