# References for higher descriptive set theory surveys

A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and similarly for the higher Cantor space $2^\kappa$, with the initial segment topology, as defined here and here. This leads (see linked questions) to many elementary questions whose answers are likely to be known.

We would appreciate references to surveys of this type of "higher descriptive set theory" and its basic results. In particular, it would be nice if these deal with results like Cantor-Bendixon Theorem, and combinatorial cardinals of the (higher) continuum.

• I remember Sy Friedman talking about descriptive-set-theoretic results for $2^\kappa$ and $\kappa^\kappa$ at the Young Set Theory meeting in 2013. This seems to be the kind of thing you're looking for. You can find slides here, and a paper here. I haven't gone through either of these very thoroughly, though. – Paul McKenney Mar 31 '16 at 19:34
• @PaulMcKenney: Thanks, this is useful! It would be very useful if someone could locate, in addition, a survey or at least a wide scope paper dealing with higher combinatorial cardinals of the (higher) continuum, such as $\mathfrak{b}_\kappa$ and $\mathfrak{d}_\kappa$. Also, a survey of more "classical" descriptive set theory ( a "higher Kechris book", so to speak) for the higher continuum is much in need. – Boaz Tsaban Mar 31 '16 at 20:08
• If by the Cantor-Bendixon theorem you mean the fact that every closed subspace of $2^{\omega}$ can be partitioned into the union of a countable scattered part and a (if non-empty) continuum sized perfect part, this was generalized to the $2^{\omega_1}$ context (the $\omega_1$-box topology) by Väänänen (A Cantor-Bendixon Theorem for the Space $\omega_1^{\omega_1}$", Fundamenta Mathematicae 1991). He uses a measurable cardinal for his consistency, but in fact you can get the consistency of the same result, and for any other particular regular uncountable $\kappa$, from just an inaccessible. – Geoff Galgon Apr 1 '16 at 18:41

I would like suggest the following three course notes given by Sy Friedman:

1) Invariant descriptive set theory (on classical and generalised Baire space),

2) Cardinal characteristics of the uncountable,

3) The higher descriptive set theory of isomorphism.

Also the monograph

Generalised descriptive set theory and classification theory

by Sy Friedman, Tapani Hyttinen and Vadim Kulikov must be very useful, in particular its introduction gives a survey of previously works on the subject.