On Borel equivalence relations in generalized Baire space

We construct two Borel equivalence relations on the generalized Baire space κ, κ = κ > ω, with the property that neither of them is Borel reducible to the other. A small modification of the construction shows that the straightforward generalization of the Glimm-Effros dichotomy fails. By λ we denote the set of all functions κ→ λ. We define a topology to (λ) by letting the sets N(η1...,ηn) = {(f1, . . . , fn) ∈ (λ)| ηi ⊆ fi for all 1 ≤ i ≤ n}, be the basic open sets, where for some α < κ for all 1 ≤ i ≤ n, ηi is a function from α to λ. We write Nη for N(η). For κ > ω, the spaces κ κ are called generalized Baire spaces. The study of these spaces started already in [Va] and since then many papers have been written on these, more on the history can be found from [FHK]. Most of the study of these spaces (for κ > ω) is done under the assumption that κ = κ and we make this assumption also. By closing open sets under complementation and unions of size ≤ κ, we get the class of Borel sets. A function between these spaces is Borel if the inverse image of every open set is Borel. As in the case κ = ω, a Borel function F is continuous on a co-meager set i.e. there are open and dense sets Ui, i < κ, such that F ( ⋂ i<κ Ui) is continuous, see [FHK]. Let X, Y ∈ {κ, 2} and let E ⊆ X and E ′ ⊆ Y 2 be equivalence relations. We say that E is Borel reducible to E ′ and write E ≤B E ′ if there is Borel function F : X → Y such that for all f, g ∈ X, fEg if and only if F (f)E ′F (g). We say that they are Borel bi-reducible if both E ≤B E ′ and E ′ ≤B E hold. In [FHK] these Borel reductions were studied. We were mostly interested in equivalence relations like isomorphism among (codes of) models of some firstorder theory but also some general theory was developed. And we were annoyed when we found out that we could not find Borel equivalence relations which are incomparable with respect to Borel reducibility. Let us see why one cannot just take some example from the case κ = ω and carry out a straightforward generalization. 2000 Mathematics Subject Classification. primary 03E15, secondary 03C75.