= Aleph-one-Suslin decomposition into analytic sets
{c}
{title2=$\aleph_1$-Suslin}
In <ZFC>, every $\aleph_1$-Suslin set is a union of $\aleph_1$ many <analytic sets>. A branch through a representing <tree> uses a countable sequence of countable ordinals. That sequence is bounded below $\omega_1$, so restrict the labels successively below $\alpha<\omega_1$.
Each restricted label set is countable and can be encoded by natural numbers; its projection is therefore an <analytic set>. The union of these projections is the original set. This is the $\kappa=\aleph_0$ case of <Successor-Suslin decomposition>. The proof explicitly uses the uncountable <cofinality> of $\omega_1$ and countable recoding, hypotheses that need care without the <axiom of choice>.
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