Corollary 41.5 (Invariance of domain):

Suppose $U$ and $V$ are homeomorphic subsets of $\mathbb{R}^n$. Then $U$ is open if and only if $V$ is open. In particular if $h:A\longrightarrow B$ is a homeomorphism between subsets of $\mathbb{R}^n$ then $h$ maps interior points of $A$ to interior points of $B$.