On Borel reducibility in generalized Baire space

Abstract

We show that although the Galvin-Prikry Theorem does not hold on generalized Baire space with the standard topol-ogy, there are similar theorems which do hold on generalized Baire space with certain coarser topologies.

Questions on generalised Baire spaces

“…If the answer is “yes”, then a partial answer to Question is obtained (cf. ): if T 1 is classifiable and shallow, T 2 is non‐classifiable and where , then .…”

“…Further background. If the answer is "yes", then a partial answer to Question 3.45 is obtained (cf [23]…”

Reducibility of Equivalence Relations Arising from Nonstationary Ideals under Large Cardinal Assumptions

“…It was asked in Friedman, Hyttinen, and Kulikov [3] and in Khomskii, Laguzzi, Löwe, and Sharankou [8,Question 3.46] whether or not the equivalence relation modulo the nonstationary ideal on the Baire space can be reduced to the Cantor space for some fixed cofinality: in our notation, whether or not E Ä;Ä S Ä Ä E 2;Ä S Ä . We approach the problem by proving several results in this direction.…”

On the reducibility of isomorphism relations

“…In , the relation , , has been studied on the closed subspaces , with and the relative subspace topology. The relation on the subspace is defined as: we say that are equivalent () if the set contains a μ‐club.…”

“…Also it was shown that if a theory has this property and , then is ‐complete. Finally, by combining [, Corollary 15] and the proof of [, Theorem 16], it follows that if T is classifiable and shallow, then is reducible to .…”