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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a normed linear space such that the countable infinite product of is homeomorphic to a normed linear space. (This is true for all Hilbert spaces, for example.) Let denote the unit sphere in . We prove the following

Theorem 1. *There is a countable cover of of open sets each of which contains no pair of antipodal points*.

Theorem 2. *There is a countable collection of closed sets in the union of which contains exactly one member of each pair of antipodal points*.

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

**[1]**C. Bessaga,*On topological classification of complete linear metric spaces*, Fund. Math.**56**(1964/1965), 251–288. MR**0178322****[2]**C. Bessaga,*Topological equivalence of unseparable reflexive Banach spaces. Ordinal resolutions of identity and monotone bases*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**15**(1967), 397–399 (English, with Loose Russian summary). MR**0221269****[3]**Czeslaw Bessaga and Victor Klee,*Two topological properties of topological linear spaces*, Israel J. Math.**2**(1964), 211–220. MR**0180825****[4]**C. Bessaga and A. Pełczyński,*Some remarks on homeomorphisms of Banach spaces*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**8**(1960), 757–761 (English, with Russian summary). MR**0132385****[5]**William H. Cutler,*Negligible subsets of infinite-dimensional Fréchet manifolds*, Proc. Amer. Math. Soc.**23**(1969), 668–675. MR**0248883**, 10.1090/S0002-9939-1969-0248883-5**[6]**David W. Henderson and R. Schori,*Topological classification of infinite dimensional manifolds by homotopy type*, Bull. Amer. Math. Soc.**76**(1970), 121–124. MR**0251749**, 10.1090/S0002-9904-1970-12392-8**[7]**V. Klee,*Mappings into normed linear spaces*, Fund. Math.**49**(1960/1961), 25–34. MR**0126690****[8]**A. P. Lundell and Stephen Weingram,*The topology of CW complexes*, Van Nostrand Reinhold, New York, 1969.**[9]**George W. Whitehead,*Homotopy theory*, Mathematics Department, Massachusetts Institute of Technology, Cambridge, Mass., 1953. Compiled by Robert J. Aumann. MR**0091469**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58B05,
57A20

Retrieve articles in all journals with MSC: 58B05, 57A20

Additional Information

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