Coverings of infinite-dimensional spheres
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- by William H. Cutler PDF
- Proc. Amer. Math. Soc. 38 (1973), 653-656 Request permission
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.References
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Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 38 (1973), 653-656
- MSC: Primary 58B05; Secondary 57A20
- DOI: https://doi.org/10.1090/S0002-9939-1973-0319221-5
- MathSciNet review: 0319221