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Tuesday, October 1, 2019

**Abstract:** Assume the axiom of determinacy. In this talk, we will discuss an almost everywhere uniformization for club subsets of $\omega_1$. Using this uniformization, we will show that every function $\Phi : [\omega_1]^{\omega_1} \rightarrow \omega_1$ has a club $C \subseteq \omega_1$ so that $\Phi | [C]^{\omega_1}$ is a continuous function. This continuity result can be used to make the following cardinality distinction, $|[\omega_1]^{<\omega_1}| < |[\omega_1]^{\omega_1}|$. This is joint work with Stephen Jackson.