Minimal $T_0$-topologies Let $X$ be an infinite set and let $\tau$ be a $T_0$-topology on $X$. Does $\tau$ contain a $T_0$-topology that is minimal with respect to $\subseteq$?

Not necessarily. See

K.-H. Pahk, Note on the characterizations of minimal $T_0$ and $T_D$ spaces, Kyungpook Math. J. 8 (1968), 5–10.

and

R.E. Larson, Minimal $T_0$-spaces and maximal $T_D$-spaces, Pacific J. Math. 31 (1969), 451–458.

