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 $\subseteq$?

Larson’s example in the paper cited by Tomek Kania and its verification are simple enough to be worth giving here (in very slightly modified form) for easy reference.

Let $\tau$ be the cofinite topology on an uncountable set $X$, let $\tau_0\subseteq\tau$ be a $T_0$ topology on $X$, and let $\tau_0^*=\tau_0\setminus\{\varnothing\}$. For each $U\in\tau_0^*$ let $\tau_U=\{W\in\tau_0:U\subseteq W\text{ or }W\subseteq U\}$; clearly $\tau_U$ is a $T_0$ topology on $X$. If $\tau_0$ is minimal $T_0$, then $\tau_U=\tau_0$ for each $U\in\tau_0^*$, and $\langle\tau_0^*,\subseteq\rangle$ is a chain. But then $\langle\{X\setminus U:U\in\tau_0^*\},\subseteq\rangle$ is an uncountable chain of finite sets, which is absurd. Thus, $\tau$ contains no minimal $T_0$ topology.

