How to proof that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic?

Hint. If $f \colon \def\R{\mathbf R}\R \to \R^2$ were a homeomorphism, what does this imply for the restriction $f\colon \R \setminus \{a\} \to \R^2 \setminus\{f(a)\}$?

Hint: Any point in $\mathbb R$ is a cut-point. While if you remove a point in $\mathbb R^2$, it remains connected because is homeomorphic to $S^1 \times \mathbb R$.