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 ${\omega }_1$. We prove, for example, that if $S$ and $T$ are stationary subsets of ${\omega }_1$ with $S\mathrm{\Delta }T=\left(S-T\right)\cup \left(T-S\right)$ stationary, then $S$ and $T$ cannot be homeomorphic. Because Ulam matrices provide ${\omega }_1$-many pairwise disjoint stationary subsets of any given stationary set, it follows that there are $2^{{\omega }_1}$-many stationary subsets of any stationary subset of ${\omega }_1$ 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 $S\times T$ is normal if and only if $S\cap T$ is stationary. In addition, we prove that for any $X\subseteq {\omega }_1$, $X\times X$ is normal, and that if $X\times X$ is hereditarily normal, then $X\times X$ is metrizable.