TITLE: Are injective continuous functions on open sets open?
QUESTION [8 upvotes]: Let $U$ be an open subset of $\mathbb{R}^n$, is there a continuous injective function $f$ from $U$ to $\mathbb{R}^n$ whose image is not open?

REPLY [10 votes]: The injective continuous image of an open $U\subseteq\mathbb{R}^n$ in $\mathbb{R}^n$ is always open - this is a consequence of invariance of domain.