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Note on "Paracompactness in Small Products"

Published online by Cambridge University Press:  20 November 2018

James Chew
James Chew*
Affiliation:
University of Akron, Akron, Ohio
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In [1], Willard proves the following

If a regular paracompact space X has a dense Lindelöfsubspace, then X is Lindelöf.

Willard notes that the above is a generalization of the standard theorem: A separable paracompact space is Lindelöf. Actually, it is a standard fact ([2, p. 24]) that a separable metacompact space is Lindelöf. Moreover, one discovers that if a separable space X is such that each open cover of X has a point-countable open refinement, then X is Lindelöf.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 4 , December 1973 , pp. 595 - 596
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Willard, S., Paracompactness in small products, Canad. Math. Bull. 14 (1971), p. 127.Google Scholar
2. Steen, and Seebach, , Counterexamples in topology, Holt, New York, 1970.Google Scholar
3. Comfort, , Hindman, and Negropontis, , F'-spaces and their product with P-spaces, Pacific J. Math. 28 (1969), 489–502.Google Scholar