real analysis - Let $Usubset mathbb{R}^n$ (open set) and $f:Ulongrightarrow V$ a homeomorphism then we can say that $V$ is a open set in $mathbb{R}^n,?$

Let $U\subset \mathbb{R}^n$ (open set) , $V\subset \mathbb{R}^n$ and $f:U\longrightarrow V$ a homeomorphism then we can say that $V$ is a open set in $\mathbb{R}^n$ ?

Any hints would be appreciated.

Answer

Yes, this result goes by the name Invariance of Domain and is due to Brouwer. It's a non-trivial result.