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.
The lattice of topologies on a set has been extensively studied; Googling on the phrase "the lattice of topologies" (with quotes) will turn up numerous references. A.K. Steiner, The lattice of topologies: structure and complementation, available here, and the references therein might be a place to start. C. Good and D.W. McIntyre, Finite intervals in the lattice of topologies, available here, has some useful references and might well be of interest in its own right.