If $f$ is a homeomorphism from an open subset $U$ of $R^n$ to a subset $V$ of $R^n$ If $f$ is a homeomorphism from an open subset $U$ of $R^n$ to a subset $V$ of $R^n$, is $V$ also open in $R^n$?
Yes, this is Brouwer's Invariance of Domain Theorem.
Edited to add (thanks commenters): if the domain of $f$ is only $U$ then you do in fact need the fancy theorem. I read the question too fast and assumed $f$ was a homeomorphism from $\Bbb{R}^2$ to itself. I will leave this answer because the comments on it are valuable.

It needs nothing as fancy as a big named theorem. A homeomorphism is a continuous bijection with continuous inverse. A function $f$ is continuous if and only if the preimage of an open set is always open. So you have these facts:

$f^{-1}$ is continuous, and

$V$ is the preimage of an open set $U$ under $f^{-1}$.

Thus $V$ is open.