Topological homogeneity and infinite powers

This research project is in general topology, and it is related to set theory in the following ways. The same could be said of my research so far. In fact, this project is the natural continuation of it. -Use set-theoretic axioms (like Martin`s Axiom or Diamond) or assumptions on cardinal invariants to prove consistency or independence results about topological statements. -Study classical combinatorial objects on omega (like filters or independent families) from the topological point of view. -Make use/investigate topological properties of definable sets (Borel, analytic, coanalytic, and so on). The above points should make clear why the Kurt Gdel Research Center for Mathematical Logic would be an appropriate host for this project: I could take advantage of the expertise that my co-applicant Sy Friedman and many other members of the KGRC (or of the neighboring logic group at TU Wien) have on the above topics, through discussion and collaboration with them. The following are the main problems that we plan to work on. -Characterize the zero-dimensional separable metrizable spaces X whose infinite power (with the usual product topology) is countable dense homogeneous (this problem is due to Fitzpatrick and Zhou). -Show that the infinite power of X is h-homogeneous whenever X is a zero-dimensional first-countable space (this problem is due to Terada). -Determine the relationships between perfect set properties in separable metrizable spaces that are not necessarily Polish (this problem is due to the applicant). -Determine for which suitable pairs of cardinals (kappa,lambda) it is possible to construct in ZFC a subset of the Cantor set with the (kappa,lambda)-Grinzing property (this problem is due to the applicant). -Investigate the relations between perfect set-type properties for ultrafilters, such as Marczewski measurability and the perfect set property for (relatively) analytic subsets (this problem is inspired by an article of Miller). -Investigate the existence of completely Baire combinatorial objects on omega, especially independent families (this problem is due to Kunen, the applicant and Zdomskyy).

In the article Non-meager free sets and independent families`` (joint work with D. Repovs and L. Zdomskyy) we showed that there exists an independent family, consisting of subsets of the natural numbers, that is as big as possible in a precise sense (the sense of Baire category). For this, we obtained a general theorem that complements a classical result of Kuratowski. In the article On Borel semifilters``, I showed that every homogeneous and definable subspace of the Cantor set (with a few trivial exceptions) has in fact a hidden combinatorial structure (more precisely, it a so-called semifilter). In the article Every filter is homeomorphic to its square`` (joint work with L. Zdomskyy), we showed that from the purely combinatorial property of being a filter, one can deduce that, from the topological point of view, nothing changes when one takes the square. This generalizes and gives an elementary proof of a result of van Engelen. In the article Infinite powers and Cohen reals`` (joint work with J. van Mill and L. Zdomskyy), we apply the method of forcing to obtain a space whose infinite power is as rigid as possible from the topological point of view. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada.