# Rodrigo Hernandez: Countable dense homogeneous spaces

20/September/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Rodrigo Hernandez (York)

Title:  Countable dense homogeneous spaces

Abstract: A separable space X is countable dense homogeneous (CDH) if every time D and E are countable dense subsets of X, there exists a homeomorphism $h:X\to X$ such that $h[D]=E$. The first examples of CDH spaces were Polish spaces. So the natural open question was whether there exists a CDH metrizable space that is not Polish. By a characterization result by Hrusak and Zamora-Aviles, such a space must be non Borel. In this talk, we will focus on recent progress in this direction. In fact, we only know about two types of CDH non-Borel spaces: non-meager P-filters (with the Cantor set topology) and $\lambda$-sets. Moreover, by arguments similar to those used for the CDH $\lambda$-set, it has also been possible to construct a compact CDH space of uncountable weight.