Compact Spaces of the First Baire Class That Have Open Finite Degree

Abstract

Abstract.We introduce the open degree of a compact space, and we show that for every natural number n, the separable Rosenthal compact spaces of degree n have a finite basis.

Fréchet Borel ideals with Borel orthogonal

“…Hence, P is infinite. This fact allows us to pick a ∈ P and s ∈ N ψ(n+1) satisfying (4). Finally, we put h(ψ(n + 1)) = d n+1 = s and g(d n+1 ) = a.…”

“…For example, Todorčević [22,Corollary 7.52] showed that I f is a selective ideal (see the definition in §2), which is the combinatorial counterpart of the fact that K is bisequential. In addition, Todorčević and Avilés in [4] gave a description of some special classes of separable Rosenthal compacta in terms of a combinatorial structure called strong n-gaps [2], which is a generalization of pairs of the form (I, I ⊥ ). In a more general setting, Krawczyk [13] and Todorčević [22,23] have shown that if I is a selective analytic ideal not countably generated, then I ⊥ is a complete co-analytic set (see also [8]).…”