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Proceedings of the American Mathematical Society

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

 

 

Cauchy sequences in semimetric spaces


Author: Dennis K. Burke
Journal: Proc. Amer. Math. Soc. 33 (1972), 161-164
MSC: Primary 54.30
DOI: https://doi.org/10.1090/S0002-9939-1972-0290328-3
MathSciNet review: 0290328
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Abstract: As the main result we prove that every semimetrizable space has a semimetric for which every convergent sequence has a Cauchy subsequence. This result is used to show that a $ {T_1}$ space X is semimetrizable if and only if it is a pseudo-open $ \pi $-image of a metric space.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0290328-3
Keywords: Semimetric space, symmetric space, Cauchy sequence, weak condition of Cauchy, developable space, pseudo open map, $ \pi $-map
Article copyright: © Copyright 1972 American Mathematical Society