Is the image of a injective immersion that is a homeomorphism onto its image from an open set, open?

Question

Let $f:U\to \mathbb{R}^n$ be a $C^1$ map from a connected open set $U$ of $\mathbb{R}^k$ where $1\leq k \leq n$ into $\mathbb{R}^n$ such that :

• $f$ is an injective immersion,
• $f$ is an homeomorphism from $U$ onto its image $f(U)$.

Does this imply that $f(U)$ is an open set of $\mathbb{R}^n$ ?

Answer 1

if you simply include the x-axis into $R^2$ it seems to me that you get a counterexample.