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ON EQUIVALENCE RELATIONS INDUCED BY LOCALLY COMPACT ABELIAN POLISH GROUPS
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 07 June 2023, pp. 1-16
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- Article
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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 
$E({\mathbb {T}}^n)$.
