Nice decompositions of R n entirely into nice sets are mostly impossible

While there are several interesting examples of partitions of R ³ into elements which individually are geometrically nice — circles or segments — the partitions themselves fail to be nice, in the sense of forming continuous or upper semicontinuous decompositions. We show that this is no accident: R ³ has no continuous decomposition into circles, and no open subset of R ⁿ has an upper semicontinuous decomposition into convex compact nonsingleton sets.