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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Perfect pre-images of collectionwise normal spaces


Author: Peg Daniels
Journal: Proc. Amer. Math. Soc. 97 (1986), 177-183
MSC: Primary 54C10; Secondary 54D15
DOI: https://doi.org/10.1090/S0002-9939-1986-0831409-2
MathSciNet review: 831409
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Abstract: We show that the following are true for perfect maps: (1) collectionwise normality with respect to compact (Lindelöf) sets is preserved inversely; (2) collectionwise normality with respect to countably compact ( $ {\omega _1}$-compact) sets is preserved inversely if and only if the domain space is normal with respect to countably compact ( $ {\omega _1}$-compact) sets; and (3) if $ P$ is any property such that (i) $ P$ is preserved by perfect maps, (ii) the free union of spaces satisfying $ P$ also satisfies $ P$, (iii) $ P$ is closed hereditary, and (iv) $ P$ plus collectionwise normality implies countable metacompactness, then collectionwise normality with respect to closed $ P$-sets is preserved inversely if the domain space is normal with respect to closed $ P$-sets. Examples of such a property $ P$ are paracompactness, submetacompactness, stratifiability and countable metacompactness.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0831409-2
Article copyright: © Copyright 1986 American Mathematical Society

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