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General embedding properties of absolute Borel and Souslin spaces


Author: Roger W. Hansell
Journal: Proc. Amer. Math. Soc. 27 (1971), 343-352
MSC: Primary 54.53; Secondary 05.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0268853-X
MathSciNet review: 0268853
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Abstract: In a recent paper, S. Willard established several characterizations of absolute metric $ {G_\alpha }$-spaces in terms of the Borel character they possessed as subspaces of certain compact Hausdorff spaces; and he asks whether a similar result holds for the $ {F_\alpha }$-spaces. In the present paper, we show that for a metric space $ X$ the following are equivalent for $ \alpha \geqq 2$ : (1) $ X$ is an absolute metric $ {F_\alpha }$-space, (2) $ X$ is a $ {Z_\alpha } \bigcap {G_\delta }$-set (i.e., a Baire set of class $ \alpha $ intersected with a $ {G_\delta }$-set) in some compactification, (3) $ X$ is an $ {F_\alpha } \bigcap {G_\delta }$-set in every completely regular Hausdorff embedding, (4) $ X$ is an absolute $ {F_\alpha }$-space with respect to the class of all perfectly normal spaces. These properties remain equivalent when `` $ {F_\alpha }$'' and `` $ {Z_\alpha }$'' are replaced by ``Souslin.'' Necessary and sufficient conditions for a metric space to be an $ {F_\alpha }$-set in all its compactifications are found and, throughout, extensions to spaces which are not necessarily metrisable are provided.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0268853-X
Keywords: Descriptive set theory, Borel classifications, Souslin schemes, absolute $ {F_\alpha }$-spaces, absolute Souslin spaces, compactifications, metric spaces
Article copyright: © Copyright 1971 American Mathematical Society

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