Topological properties of Souslin subsets

Author:
R. W. Hansell

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a subparacompact regular space such that the projection map , where is the space of irrational numbers, preserves collections of sets having a -locally finite refinement. It is shown that then preserves generalized -sets. It follows that, if has any tpological property which is hereditary with respect to generalized -sets, then every Souslin subset of 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 is a -space, thus, in particular, when 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, -embeddedness, and the properties of being a -space or -space, are also considered.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0816314-4

Keywords:
Souslin set,
generalized -set,
covering properties,
-space,
-space,
refinement -locally finite maps

Article copyright:
© Copyright 1986
American Mathematical Society