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

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  • $\begingroup$ Yes. One can make a non T_0 space "more T_0" by adding a singleton set as part of the basis, which is later removed by the intersection. Gerhard "To T_0 Or Not T_0..." Paseman, 2016.10.26. $\endgroup$ – Gerhard Paseman Oct 26 '16 at 15:18
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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.

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