Abstract and Applied Analysis
Volume 2005 (2005), Issue 3, Pages 207-219
doi:10.1155/AAA.2005.207
Local inverses of Borel homomorphisms and analytic P-ideals
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana 61801, IL, USA
Received 25 July 2004
Copyright © 2005 Sławomir Solecki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We present a theorem on the existence of local continuous
homomorphic inverses of surjective Borel homomorphisms with
countable kernels from Borel groups onto Polish groups. We also
associate in a canonical way subgroups of ℝ with
certain analytic P-ideals of subsets of ℕ. These
groups, with appropriate topologies, provide examples of Polish,
nonlocally compact, totally disconnected groups for which global
continuous homomorphic inverses exist in the situation described
above. The method of producing these groups generalizes
constructions of Stevens and Hjorth and, just as those
constructions, yields examples of Polish groups which are totally
disconnected and yet are generated by each neighborhood of the
identity.