= Successor-Suslin decomposition
{c}
In <ZFC>, if $\kappa$ is an infinite <cardinal number>, every $\kappa^+$-Suslin set is a union of $\kappa^+$ many <Kappa-Suslin sets>. Let a <tree> $T$ on $\kappa^+\times\omega$ represent the set. Each branch uses only countably many ordinal labels, so its labels are bounded below $\kappa^+$, since the <successor cardinal> is regular in <ZFC>.
For each $\alpha<\kappa^+$ restrict $T$ to labels below $\alpha$. Since $|\alpha|\leq\kappa$, its projection is a <Kappa-Suslin set>; taking their union recovers the original projection. The boundedness step requires $\operatorname{cf}(\kappa^+)>\omega$. It must not be used without a suitable choice or cofinality assumption in settings involving the <axiom of determinacy>.
Back to article page
