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

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.

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.