THE REALCOMPACTNESS NUMBER IN COUNTABLE (CROWDED) SPACES

Given a realcompact space X, we denote by Exp(X) the smallest infinite cardinal κ such that X is homeomorphic to a closed subspace of Rκ. In this talk, we analyze the realcompactness number of countable spaces. We will show that, for every cardinal κ, there exists a countable crowded space X such that Exp(X) = κ if and only if p ≤ κ ≤ c. On the other hand, we show that a scattered space of weight κ has pseudocharacter at most κ in any compactification. This will allow us to calculate Exp(X) for an arbitrary (that is, not necessarily crowded) countable space. This is a joint work with Andrea Medini and Lyubomyr Zdomskyy.