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Proceedings of the American Mathematical Society

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

 

 

Cauchy conditions on symmetrics


Author: S. W. Davis
Journal: Proc. Amer. Math. Soc. 86 (1982), 349-352
MSC: Primary 54D55; Secondary 54D20, 54E20, 54E25
DOI: https://doi.org/10.1090/S0002-9939-1982-0667305-X
MathSciNet review: 667305
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Abstract: We call a symmetric $ d$ on a space $ X$ a $ {\text{wC}}$ symmetric if whenever $ A \subseteq X$ and there exists $ \varepsilon > 0$ such that $ d(x,y) \geqslant \varepsilon $ for all $ x$, $ y \in A$, then $ A$ is relatively discrete. We show that there are no $ L$-spaces which admit $ {\text{wC}}$ symmetries. The $ {\text{wC}}$ notion is extended to certain weaker structures such as $ \mathcal{F}$-spaces with similar results.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0667305-X
Article copyright: © Copyright 1982 American Mathematical Society