Homogeneously Suslin set

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In descriptive set theory, a set $S$ is said to be homogeneously Suslin if it is the projection of a homogeneous tree. $S$ is said to be $κ$ -homogeneously Suslin if it is the projection of a $κ$ -homogeneous tree.

It can be shown that if $A\subseteq{}^\omega\omega$ is a $\mathbf{\Pi}_1^1$ set and $κ$ is a measurable cardinal, then $A$ is $κ$ -homogeneously Suslin. This result is important in the proof that the existence of a measurable cardinal implies that $\mathbf{\Pi}_1^1$ sets are determined.