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.