I have a question about the image of a homeomorphism ...
If $U\subseteq \mathbb{R}^{n}$ is open and $f: U \rightarrow f(U)\subseteq \mathbb{R}^{n}$ is a homeomorphism, then necessarily $f(U)$ is open in $\mathbb{R}^{n}$ ? Everything comes from the perturbation of the identity by a contraction, for in this case $f(U)$ is open in $\mathbb{R}^{n}$. I would appreciate to give me light on this question