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October 2009 Uncountably Many Mutually Disjoint, Dense and Convex Subsets of $\ell^2$ with Applications to Path Connected Subsets of Spheres
Samuel H. Creswell
Missouri J. Math. Sci. 21(3): 163-174 (October 2009). DOI: 10.35834/mjms/1316024882

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 $S$.

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Samuel H. Creswell. "Uncountably Many Mutually Disjoint, Dense and Convex Subsets of $\ell^2$ with Applications to Path Connected Subsets of Spheres." Missouri J. Math. Sci. 21 (3) 163 - 174, October 2009. https://doi.org/10.35834/mjms/1316024882

Information

Published: October 2009
First available in Project Euclid: 14 September 2011

zbMATH: 1189.46015
MathSciNet: MR2584542
Digital Object Identifier: 10.35834/mjms/1316024882

Subjects:
Primary: 46A55
Secondary: 52A07

Rights: Copyright © 2009 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.21 • No. 3 • October 2009
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