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Decompositions of \(\mathbb R^n,n\geq 4\), into convex sets generate codimension 1 manifold factors. (English) Zbl 1282.57028

Summary: We show that if \(G\) is an upper semicontinuous decomposition \(\mathbb R^n,n\geq 4\), into convex sets, then the quotient space \(\mathbb R^n/G\) is a codimension 1 manifold factor. In particular, we show that \(\mathbb R^n/G\) has the disjoint arc-disk property.

MSC:

57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57N75 General position and transversality
57P99 Generalized manifolds
53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
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