Minimal $T_0$-spaces

Question

As suggested by Joseph van Name in Is the associated order of a minimal $T_0$ space always total?, here's a natural question on $T_0$-spaces:

If $(X,\tau)$ is $T_0$, is there a minimal $T_0$ topology $\sigma$ on $X$ such that $\sigma\subseteq \tau$?

Answer 1

Let $X$ be an uncountable set with the cofinite topology. Then I claim there is no coarser minimal $T_0$ topology. Indeed, by Joseph Van Name's answer to the previous question, any minimal $T_0$ topology must be the minimal topology for some total specialization order. But in any total ordering of an uncountable set, there must be points with infinitely many predecessors, and so any minimal $T_0$ topology on $X$ must contain a coinfinite set.