Uncountably Many Mutually Disjoint, Dense and Convex Subsets of $\ell^2$ with Applications to Path Connected Subsets of Spheres

Abstract

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 $S$.