# Do closed subsets of the generalised Cantor space have an analogue of the perfect set property?

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.

• The answers posted to this question seem relevant to your question as well: mathoverflow.net/questions/181940/…. Jun 7 at 12:02
• Beatrice Pitton (who does not have a mathoverflow account) suggests that this paper should be relevant Jun 7 at 15:09