Seven characterizations of non-meager P-filters

We give several topological/combinatorial conditions that, for a filter on $\omega$, are equivalent to being a non-meager $\mathsf{P}$-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager $\mathsf{P}$-filter. Here, we identify a filter with a subspace of $2^\omega$ through characteristic functions. Along the way, we generalize to non-meager $\mathsf{P}$-filters a result of Miller about $\mathsf{P}$-points, and we employ and give a new proof of results of Marciszewski. We also employ a theorem of Hern\'andez-Guti\'errez and Hru\v{s}\'ak, and answer two questions that they posed. Our result also resolves several issues raised by Medini and Milovich, and proves false one "theorem" of theirs. Furthermore, we show that the statement "Every non-meager filter contains a non-meager $\mathsf{P}$-subfilter" is independent of $\mathsf{ZFC}$ (more precisely, it is a consequence of $\mathfrak{u}<\mathfrak{g}$ and its negation is a consequence of $\Diamond$). It follows from results of Hru\v{s}\'ak and van Mill that, under $\mathfrak{u}<\mathfrak{g}$, a filter has less than $\mathfrak{c}$ types of countable dense subsets if and only if it is a non-meager $\mathsf{P}$-filter. In particular, under $\mathfrak{u}<\mathfrak{g}$, there exists an ultrafilter with $\mathfrak{c}$ types of countable dense subsets. We also show that such an ultrafilter exists under $\mathsf{MA(countable)}$.Comment: 17 page

On well-splitting posets

We introduce a class of proper posets which is preserved under countable support iterations, includes $\omega^\omega$-bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.Comment: 10 pages. This is the submitted version, but it is rather close to the accepted on

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

A Hausdorff topology $\tau$ on the bicyclic monoid with adjoined zero $\mathcal{C}^0$ is called {\em weak} if it is contained in the coarsest inverse semigroup topology on $\mathcal{C}^0$. We show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on $\mathcal{C}^0$ is isomorphic to the lattice of all shift-invariant filters on $\omega$ with an attached element $1$ endowed with the following partial order: $\mathcal{F}\leq \mathcal{G}$ iff $\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\mathcal{W}$. In particular, we proved that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type $\mathfrak{t}$