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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Two examples on preimages of metric spaces


Author: N. Noble
Journal: Proc. Amer. Math. Soc. 36 (1972), 586-590
MSC: Primary 54D50
MathSciNet review: 0310840
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Abstract: Examples are given to show that the product of a compact $ {T_1}$-space with a metric space need not be a k-space, and that the product of a countably compact Hausdorff space with a metric space need not be a quasi-k-space. These examples show that the separation assumptions cannot be omitted in the following two known results: Each Hausdorff perfect preimage of a k-space is a k-space. Each regular quasi-perfect preimage of a $ {T_1}$ quasi-k-space is a quasi-k-space. Some related results, perhaps of independent interest, are also obtained.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0310840-8
PII: S 0002-9939(1972)0310840-8
Keywords: k-spaces, quasi-k-spaces, perfect preimages, quasi-perfect preimages, M-spaces, product spaces, countably compact remainders
Article copyright: © Copyright 1972 American Mathematical Society