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Proceedings of the American Mathematical Society

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

 

 

Coverings of infinite-dimensional spheres


Author: William H. Cutler
Journal: Proc. Amer. Math. Soc. 38 (1973), 653-656
MSC: Primary 58B05; Secondary 57A20
MathSciNet review: 0319221
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Abstract: Let $ F$ be a normed linear space such that the countable infinite product of $ F$ is homeomorphic to a normed linear space. (This is true for all Hilbert spaces, for example.) Let $ S(F)$ denote the unit sphere in $ F$. We prove the following

Theorem 1. There is a countable cover of $ S(F)$ of open sets each of which contains no pair of antipodal points.

Theorem 2. There is a countable collection of closed sets in $ S(F)$ the union of which contains exactly one member of each pair of antipodal points.

Theorem 3. Let $ F$ be a Hilbert space. Then there is a countable collection of sets which cover $ S(F)$ and whose diameters are less than 2.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0319221-5
Keywords: Infinite-dimensional manifold, sphere, normed linear space, projective space
Article copyright: © Copyright 1973 American Mathematical Society