Perfect pre-images of collectionwise normal spaces
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- Proc. Amer. Math. Soc. 97 (1986), 177-183 Request permission
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.References
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Additional Information
- © Copyright 1986 American Mathematical Society
- 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