Homogeneous Borel sets of ambiguous class two

Abstract

Abstract.We describe and characterize all homogeneous subsets of the Cantor set which are both an F"s and a GSo; it turns out that there are wx such spaces.

A non-homogeneous zero-dimensional 𝑋 such that 𝑋×𝑋 is a group

“…A Wadge-minimal example is the space T = X 1 2 from [8], [5]; its Wadge class isĎ 2 (Σ 0 2 ). A minimal first category example (not quite as trivially; see [6] or [7]) is Q×T = X 1 3 , of Wadge class D 3 (Σ 0 2 ). Second, does there exist a non-homogeneous space whose square is homogeneous but not a group?…”

“…In [7] it was shown that all homogeneous spaces in ∆ 0 3 contain a countable (or, in just a few cases, a σ-compact) dense subset such that any relative Π 0 2 -set in the space which contains it is actually homeomorphic to the space. In this section we will extend this result to arbitrary homogeneous Borel sets.…”

“…In dealing with this situation, we extensively use the terminology and results from [5], sections 3.4 and 4.6 (see also [4]). …”

Non-separable h-homogeneous absolute F_{σδ}-spaces and G_{δσ}-spaces

“…Van Engelen (see [E4,Theorem 4.7]) proved that if the Cantor set C is densely homogeneous with respect to an h-homogeneous space A, then C is densely homogeneous with respect to the product Q×A. Theorem 3.4 generalize this result for the non-separable case.…”

Homogeneous Borel sets

“…Together with [1], where all homogeneous Borel sets in 2" of class A°3 were determined, this yields a complete topological classification of all zero-dimensional homogeneous absolute Borel sets. Roughly, using the inductive definition of the non-self-dual Borel Wadge classes as given by Louveau [5], we show that the Wadge class of a non-A°3 homogeneous Borel set in 2" is non-self-dual and reasonably closed (for definitions, see below); then we can apply a theorem of Steel [8] to get what we want.…”