X-Suslin set ($X$-Suslin)

A set $A\subseteq {\omega }^{\omega }$ is $X$-Suslin when $A=p[T]$ for a tree $T\subseteq (X\times \omega {)}^{<\omega }$.

A $\kappa$-Suslin set is an $X$-Suslin set with $X=\kappa$. If $X$ injects into $Y$, relabelling a Suslin representation shows that every $X$-Suslin set is $Y$-Suslin.

Every $A\subseteq {\omega }^{\omega }$ is $2^{{\aleph }_0}$-Suslin. Give each $a\in A$ its own label and use the tree of finite pairs $(a\upharpoonright n,a\upharpoonright n)$, then relabel the first coordinate by an injection into a set of size $2^{{\aleph }_0}$.

For an infinite cardinal number $\kappa$, every ${\kappa }^{+}$-Suslin set is a union of ${\kappa }^{+}$ many $\kappa$-Suslin sets. A countable sequence of ordinals below the successor cardinal ${\kappa }^{+}$ is bounded there, so restrict the representing tree successively to labels below each $\alpha <{\kappa }^{+}$.

Every ${\aleph }_1$-Suslin set is a union of ${\aleph }_1$ analytic sets. Every countable sequence of countable ordinals is bounded below ${\omega }_1$, and after restricting all labels below one countable ordinal the first-coordinate space can be recoded by $\omega$.