Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8

Lineage and corrections

Banakh and Banakh, The continuity of Darboux injections between manifolds evidence for this event

Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8 parent of this event

This event attributed to Peter L.

VibeMathed record: Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8 evidence for this event

This event attributed to GPT-5.5 Pro

Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not? evidence for this event

Wong's question on MathOverflow (2016), with the five partial answers evidence for this event

Proof of Banakh-Banakh's Problems 1.7 and 1.8 evidence for this event

Both problems are answered affirmatively, in every dimension at once: every connectedness-preserving injection from a connected closed $n$-manifold into an $n$-manifold is a homeomorphism onto a component, so in particular every Darboux self-bijection of $\mathbb S^n$ and of every closed manifold is a homeomorphism. This removes the finite-$H_1$ hypothesis of the 2020 theorem for 3-manifolds and extends it above dimension 3. Compactness of the source is essential: the companion preprint (Zenodo 10.5281/zenodo.22346412) shows the corresponding statement fails for $\mathbb R^n$, $n\ge2$. The note does not address noncompact sources, manifolds with boundary, or targets of different dimension. parent of this event