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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.

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What kind of properties are you interested in? – Qiaochu Yuan 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. – Marek Jul 21 '11 at 19:33
Actually finite topological spaces are very rich from the perspective of homotopy theory! See… . – Qiaochu Yuan Jul 21 '11 at 19:37
@Qiaochu: oh, now that's something I had no idea about. Thanks! – Marek Jul 21 '11 at 19:39
There is a MO thread on compact Hausdorff topologies in the lattice of topologies:… – Martin Sleziak Nov 27 '11 at 7:05
up vote 4 down vote accepted

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.

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Sheesh, I've completely forgotten to use google :( Going to stand in the corner... – Marek Jul 21 '11 at 20:04

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