Countable spaces, realcompactness, and the pseudointersection number

Abstract

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: $(1)$ There exists a countable crowded space $X$ such that $\mathsf{Exp}(X)=\kappa$, $(2)$ $\mathfrak{p}\le \kappa \le \mathfrak{c}$. In fact, in the case $\mathfrak{d}\le \kappa \le \mathfrak{c}$, every countable dense subspace of $2^{\kappa }$ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $\kappa$ has pseudocharacter at most $\kappa$ in any compactification. This will allow us to calculate $\mathsf{Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space.

16 pages