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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Topological properties of Souslin subsets
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by R. W. Hansell PDF
Trans. Amer. Math. Soc. 293 (1986), 613-622 Request permission

Abstract:

Let $X$ be a subparacompact regular space such that the projection map $p:X \times {\mathbf {P}} \to X$, where ${\mathbf {P}}$ is the space of irrational numbers, preserves collections of sets having a $\sigma$-locally finite refinement. It is shown that $p$ then preserves generalized ${F_\sigma }$-sets. It follows that, if $X$ has any tpological property which is hereditary with respect to generalized ${F_\sigma }$-sets, then every Souslin subset of $X$ will also have this property in the relative topology. Such topological properties include nearly all covering properties (paracompactness, metacompactness, etc.), as well as normality, collectionwise normality, and the Lindelöf property. We show that the above mapping property will hold whenever $X$ is a $P(\omega )$-space, thus, in particular, when $X$ is any Souslin (hence any Baire) subset of a compact space crossed with a metrizable space. Additional topological properties of Souslin subsets, such as topological completeness, realcompactness, $z$-embeddedness, and the properties of being a $P$-space or $\Sigma$-space, are also considered.
References
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 293 (1986), 613-622
  • MSC: Primary 54H05; Secondary 04A15, 54D15
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0816314-4
  • MathSciNet review: 816314