Perfect pre-images of collectionwise normal spaces

Author:
Peg Daniels

Journal:
Proc. Amer. Math. Soc. **97** (1986), 177-183

MSC:
Primary 54C10; Secondary 54D15

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 ( -compact) sets is preserved inversely if and only if the domain space is normal with respect to countably compact ( -compact) sets; and (3) if is any property such that (i) is preserved by perfect maps, (ii) the free union of spaces satisfying also satisfies , (iii) is closed hereditary, and (iv) plus collectionwise normality implies countable metacompactness, then collectionwise normality with respect to closed -sets is preserved inversely if the domain space is normal with respect to closed -sets. Examples of such a property 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