Countable Spaces and Realcompactness

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 $\mathbb{R}^\kappa$.

In this talk, we analyze the realcompactness number of countable spaces. We will show that, for every cardinal $\kappa$, there exists a countable crowded space $X$ such that $\mathsf{Exp}(X)=\kappa$ if and only if $\mathfrak{p}\leq\kappa\leq\mathfrak{c}$. On the other hand, 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.

This is a joint work with Andrea Medini and Lyubomyr Zdomskyy.