For a regular uncountable cardinal $\kappa$, consider $2^\kappa$ with the "less than box topology" (tree topology? Easton/Bounded support topology?) in which basic open sets are of the form $[s]=\{x\mid x\supseteq s\}$ for $s\colon\kappa\to2$ a partial function with $|\operatorname{dom}(s)|<\kappa$.
Say that a tree $T\subseteq2^{{<}\kappa}$ is $\kappa$-perfect if for all $\subseteq$-chains $\langle s_\alpha\mid\alpha<\delta\rangle$ in $T$ with $\delta<\kappa$ an ordinal, there is $t\supseteq\bigcup_{\alpha<\delta}s_\alpha$ that splits in $T$.
My question is: Do closed subsets of $2^\kappa$ satisfy the $\kappa$-Perfect Set Property? What about a 'one sided' PSP: If $T\subseteq2^{{<}\kappa}$ has $2^\kappa$ many cofinal branches then does $T$ necessarily have a $\kappa$-perfect subtree?
On terminology: I have been unsuccessful thus far in finding a typical name for the topology I described, $\kappa$-perfect trees, or the $\kappa$-PSP. If you know a more common name or more common definition for these terms I would be eager to hear them.
EDIT: In the case of $\kappa=\aleph_1$, I have been directed to [1], which provides an analogue of the Cantor-Bendixson theorem for the space ${\omega_1}^{\omega_1}$. It shows that the statement "Every closed subset of ${\omega_1}^{\omega_1}$ of cardinality at least $\aleph_2$ contains a non-empty $\omega_1$-perfect subset" is equiconsistent with the existence of an inaccessible cardinal.
[1] Väänänen, Jouko, A Cantor-Bendixson theorem for the space (\omega_ 1^{\omega_ 1}), Fundam. Math. 137, No. 3, 187-199 (1991). ZBL0732.03041.
