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.