Kurt Gödel Research Center

2013 seminar talk: Seven characterizations of non-meager P-filters

Talk held by Andrea Medini (KGRC) at the KGRC seminar on 2013-12-05.

Abstract

We will begin with an introduction to topological notions of homogeneity. For example, a space is countable dense homogeneous if for every pair $(D,E)$ of countable dense subsets of $X$ there exists a homeomorphism $h:X\longrightarrow X$ such that $h[D]=E$. Then, we will gradually move to the study of the topology of filters on $\omega$, focusing on ultrafilters and non-meager filters. Here, we identify a filter with a subspace of $2^\omega$ through characteristic functions. The following is joint work with Kenneth Kunen and Lyubomyr Zdomskyy.

Recall that a filter is a P-filter if it contains a pseudointersection of each one of its countable subsets. An ultrafilter that is a P-filter is called a P-point. While Shelah showed that it is consistent that there are no P-points, it is a long standing open problem whether it is possible to construct a non-meager P-filter in ZFC. We will give several topological/combinatorial conditions that, for a filter on $\omega$, are equivalent to being a non-meager P-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager P-filter. This answers a question of Hernández Gutiérrez and Hrušák. Along the way, we also strengthen a result of Miller.

Finally, we will show that the statement "Every non-meager filter contains a non-meager P-subfilter" is independent of ZFC (more precisely, it is a consequence of $\mathfrak{u}<\mathfrak{g}$ and its negation is a consequence of $\Diamond$). It follows from results of Hrušák and Van Mill that, under $\mathfrak{u}<\mathfrak{g}$, the only possibilities for the number of types of countable dense sets of a non-meager filter are $1$ and $\mathfrak{c}$.