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Uncountably many mutually disjoint, dense and convex subsets of \(\ell^2\) with applications to path connected subsets of spheres. (English) Zbl 1189.46015

Summary: Hyperplanes in \(\mathbb R^n\) are extended to affine subspaces of \(\ell^2\) independently of the Axiom of Choice. These affine subspaces form a set of uncountably many mutually disjoint, dense and convex subsets of \(\ell^2\). A homeomorphism maps \(\ell^2\) to the sum of these sets. Further, any sphere \(S\) in \(\ell^2\) contains an uncountable collection of mutually disjoint and path connected subsets, each of which is dense in \(S\).

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46A55 Convex sets in topological linear spaces; Choquet theory
52A07 Convex sets in topological vector spaces (aspects of convex geometry)