TITLE: Continuous bijection from $\mathbb{R}^{2} \to \mathbb{R}$
QUESTION [8 upvotes]: Can anyone give an example of a continuous bijection from $\mathbb{R}^{2} \to \mathbb{R}$

REPLY [2 votes]: It is a well known result from basic topology that a continuos injective (bijective) map $$f:X\rightarrow Y$$
from a compact space $X$ into a Hausdorff space $Y$ is closed (a homeomorphism).
If you apply this result in your case to any closed disc in $\mathbb{R}^2$ and it's image you see that your bijection is a local homeomorphism, hence a homeomorphism. Using, e.g., Chandrasekhar's reason you get a contradiction.