Suppose $A$ is an open subset of $\mathbb{R}^n$ considered as a topological subspace. Suppose $B$ is a another susbet of $\mathbb{R}^n$, and under the induced topology, is homeomorphic to $A$.
If $B$ is closed and bounded, then by the Heine-Borel theorem, $B$ is compact as a topological space so $A$ and $B$ cannot be homeomorhpic, because $A$ is not compact. But if $B$ is not bounded, what can be said about $B$, if any? Must $B$ be open, can it ever be closed (let's leave out the only case of being both open and closed–the entire ambient space $\mathbb{R}^n$), or can it be neither open nor closed?
Best Answer
A subset of $\mathbb{R}^n$ homeomorphic to an open subset of $\mathbb{R}^n$ has to be open. This is the theorem of invariance of domain.