Characterization of the Subsets of Euclidean Space which are Homeomorphic to the Space Itself
According to this previous question, a subset of $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$ if and only if it is open, contractible, and simply connected at infinity.
Note that the last condition is necessary. For example, the Whitehead manifold is a contractible open subset of $\mathbb{R}^3$, but it is not homeomorphic to $\mathbb{R}^3$.