The descriptive complexity of the set of all closed zero-dimensional subsets of a Polish space

Abstract

Given a space X we investigate the descriptive complexity class ΓX of the set F0(X) of all its closed zero-dimensional subsets, viewed as a subset of the hyperspace F(X) of all closed subsets of X. We prove that max{ΓX ; X analytic } = Σ 1 2 and sup{ΓX ; X Borel Π 0 ξ } ⊇ Σ 0 ξ for any countable ordinal ξ ≥ 1. In particular we prove that there exists a one-dimensional Polish subpace of 2 ω × R 2 for which F0(X) is not in the smallest non trivial pointclass closed under complementation and the Souslin operation A . 1 2010 Mathematics Subject Classification. Primary: 03E15, 54H05; Secondary: 54F45.

The descriptive complexity of connectedness in Polish spaces

“…Note that the graph G := Gr(f ) ≈ P is not connected. However as shown in [7] (see also [2]) iff : I → I is any extension of f withf (q) ∈ J q for all q ∈ Q then Gr(f ) is connected. In particular the closure of G in I 2 :…”

“…Also given any pointclass Γ we denote byΓ its dual class, so for any Polish space X,Γ(X) = {X \ A : A ∈ Γ}; and by A(Γ) the class of all sets obtained by operation A applied to a Souslin scheme of sets from Γ. It is well known that A(Σ 1 1 ) = Σ 1 1 and we shall consider in the sequel the class A(Π 1 1 ) and its dual classǍ(Π 1 1 ) which are both subclasses of the class of C-sets, hence of the class ∆ 1 2 (see [1] or [2] for more details).…”

“…We also need the following result from [2]. (Observe that the set F of Lemma 9.5 in [2] is the graph of a function g with the properties stated in Lemma 7.4.) Then Φ is clearly continuous and we now check that Φ reduces the Π 1 1 -complete set P 1 = P 1 (Q, ∆) toČ(X).…”

“…This work is a continuation of [2] where was investigated the maximal possible complexity of the set of all zero-dimensional closed subsets of a given Polish space. In the present work we study the analogous question for connectedness which is a kind of strong negation of zero-dimension.…”

“…We mention in particular the case of the set F 0 (X) of all closed zero-dimensional subsets of a Polish space X. It is not difficult to check that F 0 (X) is a Σ 1 2 set, and it follows from the results of [2] that the maximal complexity of this set is at least Σ 0 2 (see [8] or [2] for the definition of this class of ∆ 1 2 sets). But the maximal complexity of the set F 0 (X) is unknown.…”