 Successor-Suslin decomposition

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 $|\alpha |\le \kappa$, its projection is a Kappa-Suslin set; taking their union recovers the original projection. The boundedness step requires $\mathrm{cf}({\kappa }^{+})>\omega$. It must not be used without a suitable choice or cofinality assumption in settings involving the axiom of determinacy.