Topology inside $\omega_1$

Abstract

In this expository paper, we show how the pressing down lemma and Ulam matrices can be used to study the topology of subsets of ω1. We prove, for example, that if S and T are stationary subsets of ω1 with SΔT=(S−T)∪(T−S) stationary, then S and T cannot be homeomorphic. Because Ulam matrices provide ω1-many pairwise disjoint stationary subsets of any given stationary set, it follows that there are 2ω1-many stationary subsets of any stationary subset of ω1 with the property that no two of them are homeomorphic to each other. We also show that if S and T are stationary sets, then the product space S×T is normal if and only if S∩T is stationary. In addition, we prove that for any X⊆ω1, X×X is normal, and that if X×X is hereditarily normal, then X×X is metrizable.