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: • There exists a countable crowded space X such that $\mathsf {Exp}(X)=\kappa $. • $\mathfrak {p}\leq \kappa \leq \mathfrak {c}$. In fact, in the case $\mathfrak {d}\leq \kappa \leq \mathfrak {c}$, 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 X.