{c}
{title2=$\kappa$-Suslin}

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.

Kappa-Suslin set

Let $T\subseteq(X\times\omega)^{<\omega}$ be a <Suslin representation> of $A$, and let $f:X\to Y$ be injective. Apply $f$ coordinatewise to the first coordinate of every node and put
$$
T_f=\{(f\circ s,t):(s,t)\in T\}\subseteq(Y\times\omega)^{<\omega}.
$$
The injectivity of $f$ ensures that a sequence is a branch of $T_f$ exactly when its first coordinate decodes to a branch of $T$. Hence $p[T_f]=p[T]=A$, proving that every <X-Suslin set> is $Y$-Suslin whenever $X$ injects into $Y$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 158 / 2 / i / Solution

{c}

Let $T\subseteq(X\times\omega)^{<\omega}$ be a <tree>. Its projection is
$$
p[T]=\{x\in\omega^\omega:\exists y\in X^\omega\ ((y,x)\in[T])\}.
$$
The tree $T$ is a Suslin representation of $p[T]$.

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 Incoming links: Suslin representation