Consider a set $X$ and a set $T$ of topologies on $X$. Then $(T, \leq)$ (with $\sigma \leq \tau$ if $\sigma$ is coarser than $\tau$) forms a bounded lattice with join given by intersection and meet $\sigma \vee \tau$ given by the unique coarsest topology containing $\sigma \cup \tau$. Is there anything reasonable that can be said about this lattice? I wonder whether people have studied similar stuff and if so I'd like to see some references.
My motivation stems mainly from my playing with topologies on finite sets, so this is the case I'd be interested in the most.