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 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?…”
Section: Discussionmentioning
confidence: 99%
“…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.…”
Section: The Main Lemmamentioning
confidence: 99%
“…In dealing with this situation, we extensively use the terminology and results from [5], sections 3.4 and 4.6 (see also [4]). …”
Abstract. We provide an example of a zero-dimensional (separable metric) absolute Borel set X which is not homogeneous, but whose square X × X admits the structure of a topological group. We also construct a zero-dimensional absolute Borel set Y such that Y is a homogeneous non-group but Y × Y is a group. This answers questions of Arhangel'skiȋ and Zhou.
“…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?…”
Section: Discussionmentioning
confidence: 99%
“…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.…”
Section: The Main Lemmamentioning
confidence: 99%
“…In dealing with this situation, we extensively use the terminology and results from [5], sections 3.4 and 4.6 (see also [4]). …”
Abstract. We provide an example of a zero-dimensional (separable metric) absolute Borel set X which is not homogeneous, but whose square X × X admits the structure of a topological group. We also construct a zero-dimensional absolute Borel set Y such that Y is a homogeneous non-group but Y × Y is a group. This answers questions of Arhangel'skiȋ and Zhou.
“…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.…”
Denote by Q(k) a σ-discrete metric weight-homogeneous space of weight k. We give an internal description of the space Q(k) ω . We prove that the Baire space B(k) is densely homogeneous with respect to Q(k) ω if k > ω. Properties of some non-separable h-homogeneous absolute F σδ -sets and G δσ -sets are investigated.
“…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.…”
Abstract.Topological characterizations of all zero-dimensional homogeneous absolute Borel sets are obtained; it turns out that there are u, such spaces. We use results from game theory-particularly, about Wadge classes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.