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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

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up vote 2 down vote accepted

The Invariance of domain theorem gives you a quick answer in the positive, but there might be a more elementary approach.

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