Absolute $\mathcal F$-Borel classes

Abstract

We investigate and compare F-Borel classes and absolute F-Borel classes. We provide precise examples distinguishing these two hierarchies. We also show that for separable metrizable spaces, F-Borel classes are automatically absolute.Borel sets (in some topological space X) are the smallest family containing the open sets, which is closed under the operations of taking countable intersections, countable unions and complements (in X). The family of F -Borel (resp. G-Borel ) sets is the smallest family containing all closed (resp. open) sets, which is closed under the operations of taking countable intersections and countable unions of its elements. In metrizable spaces the families of Borel, F -Borel and G-Borel sets coincide. In non-metrizable spaces, open set might not necessarily be a countable union of closed sets, so we need to make a distinction between Borel, F -Borel and G-Borel sets.In the present paper, we investigate absolute F -Borel classes. While Borel classes are absolute by [HS] (see Proposition 1.8 below), by [Tal] it is not the case for F -Borel classes (see Theorem 1.7 below). We develop a method of estimating absolute Borel classes from above, which enables us to compute the exact complexity of Talagrand's examples and to provide further examples by modifying them. Our main results are Theorem 5.14 and Corollary 5.15.The paper is organized as follows: In the rest of the introductory section, we define the Borel, F -Borel and G-Borel hierarchies and recall some basic results. In Section 2, we recall the definitions and basic results concerning compactifications and their ordering. In Theorem 2.3, we show that the complexity of separable metrizable spaces coincides with their absolute complexity (in any of the hierarchies). Section 3 is devoted to showing that absolute complexity is inherited by closed subsets. In Section 4, we study special sets of sequences of integers -trees, sets which extend to closed discrete subsets of ω ω and the 'broom sets' introduced by Talagrand. We study in detail the hierarchy of these sets using the notion of rank. In Section 5, we introduce the class of examples of spaces used by Talagrand. We investigate in detail their absolute complexity and in Section 5.3, we prove our main results.Let us start by defining the basic notions. Throughout the paper, all the spaces will be Tychonoff. For a family of sets C, we will denote by C σ the collection of all countable unions of elements of C and by C δ the collection of all countable intersections of elements of C.Definition 1.1 (Borel classes). Let X be a topological space. We define the Borel multiplicative classes M α (X) and Borel additive classes A α (X), α < ω 1 , as follows:• M 0 (X) = A 0 (X) := the algebra generated by open subsets of X,• M α (X) := β<α (M β (X) ∪ A β (X)) δ for 1 ≤ α < ω 1 ,• A α (X) := β<α (M β (X) ∪ A β (X)) σ

Complexities and representations of $\mathcal F$-Borel spaces

“…To prove Lemma 7.9 for a broom space T contained in a topological space Y , we first need to have a Suslin scheme C in T which satisfies A(C Proof. This holds, for example, by [Kov,Lemma 5.10].…”

“…197], so it remains to prove (ii). For even α, the result is proven in [Kov,Prop. 5.13] (but the proof below also applies).…”

“…As in the proof of Proposition 1.7 (4), we can assume that α = 2. The existence of X β 2 follows from [Kov,Theorem 5.14] for even β, resp. from Corollary 7.28 for odd β.…”

“…To prove (ii) =⇒ (i), it suffices to show that (ii) implies that X is F α in every compactification (by, for example, [Kov,Remark 1.5]).…”

“…The first part of (3) is by no means obvious -we are dealing with the complexity of X in all compactifications, not only those which are metrizable. Nonetheless, it's proof is fairly elementary; see for example [Kov,Theorem 2.3]. The "in particular" part follows from the fact that the Borel hierarchy in Polish spaces is non-trivial (see, for example, the existence of universal sets in [Sri]).…”