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 \(\mathfraku<\mathfrakg\) and its negation is a consequence of \(\Diamond\)). It follows from results of Hrušák and Van Mill that, under \(\mathfraku<\mathfrakg\), the only possibilities for the number of types of countable dense sets of a non-meager filter are \(1\) and \(\mathfrakc\).