Questions on topological homogeneity

This research project is in general topology, and it is related to set theory in the following ways. -Use set-theoretic axioms (like Martins Axiom or V=L) or assumptions on cardinal invariants to prove consistency or independence results about topological statements. -Study combinatorial objects on omega (especially filters) from the topological point of view. -Make use of/investigate topological properties of definable sets (Borel, analytic, coanalytic, and so on). The focus of our research will be on various notions of homogeneity and rigidity. Intuitively, a space is homogeneous if it looks the same everywhere, while rigid spaces lie at the opposite end of the spectrum. In particular, we will consider homogeneity with respect to countable dense sets in the context of infinite powers, filters, and function spaces. We also plan to address several questions that were left open in a recent collaboration with Z. Vidnynszky. These questions involve sigma-homogeneity and strong notions of rigidity. Finally, we will investigate questions of J. van Mill regarding actions of Polish groups and Polish spaces.

We will split this summary in three parts, each one named after the title of the corresponding research article. (1) Every finite-dimensional analytic space is sigma-homogeneous (by C. Agostini, A. Medini). In 2011, Ostrovsky obtained the surprising result that every finite-dimensional Borel space is sigma-homogeneous (that is, a countable union of homogeneous subspaces). However, since his methods are heavily Wadge-theoretic, one would require determinacy assumptions in order to apply them beyond the Borel realm. In this article, we were able to circumvent this obstacle and show that Ostrovsky's result can be extended to all finite-dimensional analytic spaces without additional set-theoretic assumptions. This answers Question 8.2 from the project description. (2) Countable dense homogeneity and topological groups (by C. Agostini, A. Medini, L. Zdomskyy). A separable space X is countable dense homogeneous if for every pair (D,E) of countable dense subsets of X there exists a homeomorphism h of X such that h[D]=E. Familiar spaces like the reals, the Cantor set, and the Hilbert cube are all examples of countable dense homogeneous spaces. A long-standing theme in this area is to find non-Polish examples of countable dense homogeneous spaces. The first ZFC non-Polish example was obtained by Farah, Hrusak and Martinez Ranero in 2005, using metamathematical methods. More ``down-to-earth'' examples followed. In this article, we gave another ZFC non-Polish example that has the additional, very strong property of being a topological group. This answers Question 6.4 from the project description. (3) Countable spaces, realcompactness, and the pseudointersection number (by C. Agostini, A. Medini, L. Zdomskyy). A realcompact space X is one that is homeomorphic to a closed subspace of R^kappa for some cardinal kappa. The minimum such kappa is the realcompactness number of X. A classical result of Hechler shows that the realcompactness number of Q is the dominating number d. The main result of this article finds an unexpected connection with the pseudointersection number p, when one considers arbitrary (that is, not necessarily metrizable) countable crowded spaces instead of Q. More precisely, the following are equivalent for every cardinal kappa: - p kappa c, - There exists a countable crowded space X such that the realcompactness number of X is kappa. We remark that the results of this article grew out of the investigation of Problem 6.6 from the project description. Unfortunately, this problem remains open.