Given a cardinal $\kappa $ and filters $\mathcal {F}_\alpha $ on $\omega $ for $\alpha \in \kappa $, we will show that if $\prod _{\alpha \in \kappa }\mathcal {F}_\alpha $ is countable dense homogeneous then $\kappa <\mathfrak {p}$ and each $\mathcal {F}_\alpha $ is a non-meager $\mathsf {P}$-filter. This partially answers a question of Michael Hrušák. Along the way, we will show that the product of fewer than $\mathfrak {p}$ non-meager $\mathsf {P}$-filters has the Miller property ($\mathsf {MP}$). We will also describe explicitly the connection between Miller-measurability and the $\mathsf {MP}$. As a corollary, we will see that the intersection of fewer than $\mathsf {add}(m^0)$ non-meager $\mathsf {P}$-filters is a non-meager $\mathsf {P}$-filter, where $m^0$ denotes the ideal of Miller-null sets. We will conclude by investigating the preservation of the $\mathsf {MP}$ under intersections and products.

All spaces are assumed to be Tychonoff. Given a realcompact space X, we denote by $\mathsf {Exp}(X)$ the smallest infinite cardinal $\kappa $ such that X is homeomorphic to a closed subspace of $\mathbb {R}^\kappa $. Our main result shows that, given a cardinal $\kappa $, the following conditions are equivalent:

• • There exists a countable crowded space X such that $\mathsf {Exp}(X)=\kappa $.

• • $\mathfrak {p}\leq \kappa \leq \mathfrak {c}$.

$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$

$2^\kappa $

$\kappa $

$\kappa $

$\mathsf {Exp}(X)$

A $Q$ -set is a set of reals every subset of which is a relative ${{G}_{\delta }}$ . We investigate the combinatorics of $Q$ -sets and discuss a question of Miller and Zhou on the size $q$ of the smallest set of reals which is not a $Q$ -set. We show in particular that various natural lower bounds for $q$ are consistently strictly smaller than $q$ .