Abstract
We show that the enveloping space \({\mathbb {X}}_G\) of a partial action of a Polish group G on a Polish space \({\mathbb {X}}\) is a standard Borel space, that is to say, there is a topology \(\tau \) on \({\mathbb {X}}_G\) such that \(({\mathbb {X}}_G, \tau )\) is Polish and the quotient Borel structure on \({\mathbb {X}}_G\) is equal to \(Borel({\mathbb {X}}_G,\tau )\). To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught’s transform are valid for partial actions of groups.
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The authors thank La Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander for the financial support for this work, which is part of the VIE Project # 5761.
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Pinedo, H., Uzcategui, C. Borel globalizations of partial actions of Polish groups. Arch. Math. Logic 57, 617–627 (2018). https://doi.org/10.1007/s00153-017-0598-8
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DOI: https://doi.org/10.1007/s00153-017-0598-8