math.GN — General Topology

Recurrent tubes and connectedness-preserving bijections of Euclidean spaces

Contributed by Peter Lim

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 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.