Geometric interpretation of Lawvere-Tierney topology

Question

A Lawvere-Tierney topology $j$ on a sheaf topos $\mathcal{G}$ yelds a closure operator $\text{Sub}(1) \to \text{Sub}(1)$. If the topos is localic $\mathcal{G} \cong \text{Sh}(L)$ this yields a closure operator on the lattice of open sets $$\Gamma: \mathcal{O}(L) \to \mathcal{O}(L). $$

From a topological point of view, what is this closure operator?

Answer 1

There is a correspondence between the class of topologies on X and the class of Lawvere-Tierney topology over the topos $\text{Set}^{\mathcal{P}(X)^{\text{op}}}$, $$\{\text{topologies on } X\} \leftrightarrow \{\text{LT-topologies on } \text{Set}^{\mathcal{P}(X)^{\text{op}}}\}. $$ Here LT topologies essentially correspond to the closure operator $j :\mathcal{P}(X) \to \mathcal{P}(X)$ associated to the comonad $\tau \subset \mathcal{P}(X)$ for a topology $\tau$ on $X$.

Thus, an LT topology can be seen as a specification of a subtopology on a topological space.