Recurrent tubes and connectedness-preserving bijections of Euclidean spaces
For every \(n\ge 2\) we construct a connectedness-preserving bijection \(F\) of \(\mathbb{R}^n\) whose inverse does not preserve connectedness. The restriction of \(F\) off the closed coordinate ray \(E\) is a homeomorphism onto the complement of \(L=K\setminus\{\infty\}\), where \(K\subset\mathbb{S}^n\) is a nondegenerate continuum with empty interior. The discontinuity sets of \(F\) and \(F^{-1}\) are exactly \(E\) and \(L\), and both maps can be chosen to have exact Baire class \(2\). The preimage of a fixed straight segment can be any prescribed uncountable closed subset of \(E\) with…
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