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

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

• 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. – Gerhard Paseman Oct 26 '16 at 15:18

• 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.