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Recurrent tubes and connectedness-preserving bijections of Euclidean spaces

Primarily AI-generated textHuman understanding: some partsmath.GN — General Topology

Contributed by Peter Lim ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

For every n≥2n\ge 2 we construct a connectedness-preserving bijection FF of Rn\mathbb{R}^n whose inverse does not preserve connectedness. The restriction of FF off the closed coordinate ray EE is a homeomorphism onto the complement of L=K∖{∞}L=K\setminus\{\infty\}, where K⊂SnK\subset\mathbb{S}^n is a nondegenerate continuum with empty interior. The discontinuity sets of FF and F−1F^{-1} are exactly EE and LL, and both maps can be chosen to have exact Baire class 22. The preimage of a fixed straight segment can be any prescribed uncountable closed subset of EE with empty relative interior, including the middle-thirds Cantor set. The construction is uniform in the dimension. Homeomorphisms with compact support move finitely many collar points to a prescribed net, and narrowing the resulting collars produces a recurrent tube. Compact arcs on one retained boundary line supply dense connected tracks, while a gate lemma proves preservation of arbitrary connected sets. This answers W. Wong's question in the negative. We also prove that every component of the closure of the discontinuity set of a planar connectedness-preserving bijection is unbounded, and construct Borel counterexamples on manifolds with a standard spherical end.

Provenance statement

The manuscript has been read by Willie Wong, the person who posed the question that the manuscript answers. It is believed to be "likely correct"

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Version history

  1. v1Submitted by Peter LimInitial depositCurrentOct 25, 2026