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ON EQUIVALENCE RELATIONS INDUCED BY LOCALLY COMPACT ABELIAN POLISH GROUPS
Published online by Cambridge University Press: 07 June 2023
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
$E({\mathbb {T}}^n)$
.
MSC classification
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- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
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