March 7
Maya Saran,
Mathematics Foundation of America
A descriptive-set-theoretic result on sigma-ideals of compact sets
Polish spaces, the objects of study of descriptive set theory, are completely metrizable topological spaces that have a countable dense subset. For example, the reals - the first Polish space in the world. We will look at 'sigma-ideals' of compact subsets of a Polish space. Think of a sigma-ideal as being a collection of 'small' compact sets, under some notion of smallness -- so for example, your Polish space could be the interval $[0,1]$ and your sigma-ideal could be the collection of all its compact sets of Lebesgue measure $0$. The descriptive-set-theoretic study of these objects yields rich results for the following reason. If you look at the collection of all the compact subsets of a Polish space, that too, topologized and metrized in a natural way, turns out to be a Polish space. This means that you can look at your sigma-ideal of compact sets in two places: in the original space, say $E$, and in the `hyperspace' of all compact sets of $E$. In this talk we will deal with sigma-ideals that can be represented in a very nice way inside this hyperspace, and we will examine the behaviour of so-called G-delta subsets of $E$ with respect to this representation.