The continuity of Darboux injections between manifolds

TL;DR

This paper establishes conditions under which injective maps between certain connected metrizable spaces are continuous, focusing on cases where the target space is a low-dimensional manifold and the domain has specific compactness or connectivity properties.

Contribution

It provides new partial results on the continuity of Darboux injections between manifolds, addressing a problem posed by Willie Wong.

Findings

Injective maps are continuous under specified conditions involving manifold dimension and compactness.

The results extend known cases of Darboux injections to higher-dimensional manifolds.

Addresses a problem on the continuity of Darboux injections posed on Mathoverflow.

Abstract

We prove that an injective map $f:X\to Y$ between connected metrizable spaces $X,Y$ is continuous if for every connected subset $C\subset X$ the image $f(C)$ is connected and one of the following conditions is satisfied: (1) $Y$ is a 1-manifold and $X$ is compact; (2) $Y$ is a 2-manifold and $X$ is a closed $n$-manifold of dimension $n\ge 2$; (3) $Y$ is a 3-manifold and $X$ is a simply-connected closed $n$-manifold of dimension $n\ge 3$. This gives a partial answer to a problem of Willie Wong, posed on Mathoverflow.