Coverings of infinite-dimensional spheres

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.