Is every topology the intersection of the $T_0$-topologies containing it?

Question

This is a continuation of the question about Minimal $T_0$-spaces .

Let $X\neq \emptyset$ be a set and let $\text{Top}(X)$ denote the lattice of all topologies on $X$ and let $\tau\in\text{Top}(X)$.

Do we have $$\tau = \bigcap\{\sigma \in \text{Top}(X): \sigma \text{ is } T_0 \text{ and } \sigma \supseteq \tau\}?$$

Answer 1

I've answered the respective full question around 1982 or even many years earlier. Possibly, the result was well known "always", before me.

The answer is contained in one of my OM Answers from a different thread:

THEOREM

• Every topology is an intersection of singular spaces.
• Every topology $\ T\ $ is the intersection of all maximal non-discrete topologies which contain $\ T.$

And, every singular topology is $T_0,\ $ and every maximal non-discrete topology is singular.