Abstract In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by alpha $\alpha $ α -CLI and L- alpha $\alpha $ α -CLI where alpha $\alpha $ α is a countable ordinal. We establish three results: (1) G is 0 $0$ 0 -CLI iff upper G equals StartSet 1 Subscript upper G Baseline EndSet $G=\{1_G\}$ G = { 1 G } ; (2) G is 1 $1$ 1 -CLI iff G admits a compatible complete two-sided invariant metric; and (3) G is L- alpha $\alpha $ α -CLI iff G is locally alpha $\alpha $

Abstract Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^{\omega }/c(G)$ , where $c(G)$ is the group of all convergent sequences in G. The connected component of the identity of a Polish group G is denoted by $G_0$ . Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq _B E(H)$ , then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker (S)$ is non-archimedean. The converse is also true when G is connected and compact. For $n\in {\mathbb {N}}^+$ , the partially ordered set $P(\omega )/\mbox {Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb {R}}^n)$ and $E({\mathbb {T}}^n)$ .

Abstract We prove the Decomposability Conjecture for functions of Baire class $2$ from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.

Abstract In this article, a notion of Schauder equivalence relation ℝℕ/L is introduced, where L is a linear subspace of ℝℕ and the unit vectors form a Schauder basis of L. The main theorem is to show that the following conditions are equivalent: (1) the unit vector basis is boundedly complete; (2) L is a Fσ in ℝℕ; (3) ℝℕ/L is Borel reducible to ℓ∞. We show that any Schauder equivalence relation generalized by a basis of ℓ2 is Borel bireducible to ℝℕ/ℓ2 itself, but it is not true for bases of c0 or ℓ1. Furthermore, among all Schauder equivalence relations generated by sequences in c0, we find the minimum and the maximum elements with respect to Borel reducibility.

Abstract We investigate Borel reducibility between equivalence relations E(X; p) = Xℕ/ℓp(X)'s where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(Lr; p)'s and E(c0; p)'s for r, p Є [1, +∞). We also answer a problem presented by Kanovei in the affirmative by showing that C(ℝ+)/C0(ℝ+) is Borel bireducible to ℝℕ/c0.

Abstract We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility hierarchy determined.