In this expository paper, we show how the pressing down lemma and Ulam matrices can be used to study the topology of subsets of . We prove, for example, that if and are stationary subsets of with stationary, then and cannot be homeomorphic. Because Ulam matrices provide -many pairwise disjoint stationary subsets of any given stationary set, it follows that there are -many stationary subsets of any stationary subset of with the property that no two of them are homeomorphic to each other. We also show that if and are stationary sets, then the product space is normal if and only if is stationary. In addition, we prove that for any , is normal, and that if is hereditarily normal, then is metrizable.
"Topology inside $\omega_1$." Rocky Mountain J. Math. 50 (6) 1989 - 2000, December 2020. https://doi.org/10.1216/rmj.2020.50.1989