# What can be said about the lattice of topologies on a given set?

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.

• What kind of properties are you interested in? Jul 21 '11 at 19:30
• @Qiaochu: well, I'm not quite sure. I guess I would be interested in both topological properties (I am aware that finite-set topology isn't very rich, e.g. w.r.t. separation axioms) and lattice-theoretical properties (i.e. whether this lattice is somehow more special than any other random lattice). But mostly I just wondered whether anyone besides me has asked these questions before and what where the study of it can lead. Jul 21 '11 at 19:33
• Actually finite topological spaces are very rich from the perspective of homotopy theory! See mathoverflow.net/questions/45549/… . Jul 21 '11 at 19:37
• @Qiaochu: oh, now that's something I had no idea about. Thanks! Jul 21 '11 at 19:39
• In the book Kolibiar, Šalát, Legéň, Znám: Algebra a príbuzné disciplíny, two papers are mentioned as references for lattices of topologies: Larsson, Andima: The lattice of topologies and Rosický: Sublattices of the lattice of topologies, Acta Fac. Rerum Natur. Univ. Comenian. Math.Special Numbers, 1975, 39-41. I did not find the second one online - but I thought that for you it might be interesting to know that research in this area has been done by Czech mathematicians too. Nov 28 '11 at 21:04