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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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    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](http://dx.doi.org/10.1216/RMJ-1975-5-2-177) 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.2011-11-28

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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...2011-07-21