Nice decompositions of \(R^ n\) entirely into nice sets are mostly impossible (Q1922811)

This paper presents two results: (1) \(\mathbb{R}^3\) admits no continuous decomposition into (round) circles, and (2) every upper semicontinuous decomposition of an open subset of \(\mathbb{R}^n\) into compact, convex sets must involve at least one singleton. The second of these is the more interesting. The first, although neatly proved, is not new; indeed, more general results ar available. Just as in the paper at hand, one can show directly that any continuous decomposition of \(\mathbb{R}^n\) into isometric copies of the standard \(k\)-sphere would be a locally trivial fibration, but an old result of \textit{A. Borel} and \textit{J. P. Serre} [C. R. Acad. Sci., Paris 231, 943-945 (1950; Zbl 0039.19201)] established that \(\mathbb{R}^n\) could not be fibered by any compact space except a point. Furthermore, by work of \textit{J. Walsh} and the reviewer [Trans. Am. Math. Soc. 288, 273-291 (1985; Zbl 0568.57013)], \(\mathbb{R}^n\) admits no upper semicontinuous decomposition into \((n-2)\)-spheres. Also worth noting, contrary to a statement amidst the final questions: \(\mathbb{R}^n\) admits no upper semicontinuous decomposition into arcs [\textit{S. L. Jones}, Bull. Am. Math. Soc. 74, 155-159 (1968; Zbl 0171.21604)].