applications of topological spaces that are $T_1$, but are not $T_2$

Question

If a topological space is $T_1$, its specialization preorder is the equality (so it is trivial). If a topological space is $T_2$ (Hausdorff), a limit of a proper filter is unique. I suppose this is why topological spaces studied in analysis are $T_2$. It seems to me there is a gap between $T_1$ and $T_2$ axioms. Are there any interesting applications of topological spaces that are $T_1$, but are not $T_2$?