Borel extractions of converging sequences in compact sets of Borel functions

Abstract

It is well known by a classical result of Bourgain-Fremlin-Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence (f n ) n∈ω in K one can extract a subsequence (f n k ) k∈ω converging to f . In the present work we prove that this extraction can be achieved in a "Borel way." This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence. This paper is written in a double aim: firstly to prove the results announced in the abstract above, secondly to emphasize the interest of "effective" methods in the treatment of classical descriptive problems. The paper is indeed conceived as an invitation to the non specialist reader for the use of these powerful methods. And though a thorough reading of the paper necessitates clearly a great familiarity with Effective Descriptive Set Theory (for example as developed in [9]) our deep hope is that the text will be comprehensible, at least in its general framework, by a reader with a standard classical background. To that end we shall give in Section 1 a quick informal presentation of the main effective concepts and results used in this work.The basic topological properties of Rosenthal compacta, or more generally of compact spaces of Borel functions on a given Polish space, were entirely elucidated by the fundamental works of Rosenthal [10] and later on Bourgain, Fremlin and Talagrand [3]. The problems discussed in the present work concern the descriptive properties of these spaces. This was actually initiated in [4], but it is only quite recently that the interest of such a study was highlighted first by the work of Dodos in [5] and since then by several other interesting applications (see [1,2,7]). In fact our initial motivation was to solve a natural problem left open in our earlier study [4]. Incidently the solution of this old problem

Operators whose dual has non-separable range

“…We have already observed that B X * * consists only of Baire-1 functions and that the sequence (d n ) is dense in B X * * . As it was explained in [9, Remark 1(2)], by Debs' Theorem [7] (see also [8]) there exists a hereditary, Borel and cofinal subfamily F 0 of D. With the same reasoning, we see that there exists a hereditary, Borel and cofinal subfamily F 1 of R. We set F = F 0 ∩ F 1 . Clearly the family F is as desired.…”

“…By Fact 15, we see that the family D ∩ R is hereditary, co-analytic and cofinal in [N] ∞ . We will need the following stronger property which is essentially a consequence of the deep effective version of the Bourgain-Fremlin-Talagrand Theorem due to G. Debs [7,8]. Proof.…”

Ideals on countable sets: a survey with questions

“…A particular interesting example is for K a separable Rosenthal compacta. By Debs' theorem [7,8] (see also [9]), in every Rosenthal compacta, C(x n ) n is uniformly tall. When K is not first countable C(x n ) n is a complete co-analytic subset of N [∞] .…”

Fréchet Borel ideals with Borel orthogonal

“…The result of Krawczyk about Rosenthal compacta [13] says that this is not the case when the compactum is not first countable. Nevertheless, Debs [6,7] has shown that there is a Borel set that codes the convergence relation in a Rosenthal compactum. In the last section we shall see how the ideals in B can be used to construct a family of size ℵ 1 of pairwise non homeomorphic countable sequential spaces such that both the topology and the convergence relation are Borel and yet all of them have sequential order ω 1 .…”