A topological characterization of a class of cardinals

Abstract:

Let $\mathfrak {m}$ be the first measurable cardinal. We say that a cardinal $\alpha$ is Ulam-stable if, on the discrete space $D(\alpha )$ of cardinal $\alpha$, every filter with $\mathfrak {m}$-intersection property can be extended to an ultrafilter with $\mathfrak {m}$-intersection property. The main result we prove is the following: $\alpha$ is Ulam-stable if and only if its Hewitt-Nachbin realcompletion $\upsilon D(\alpha )$ is paracompact.