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 $\sigma\subseteq \tau$?

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 $\sigma\subseteq \tau$?

As suggested by Joseph van Name in Is the associated order of a minimal $T_0$ space always total?, here' 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 $\sigma\subseteq \tau$?

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 $\sigma\subseteq \tau$?