TITLE: Continuous bijections between $\mathbb{R}^2$ and $\mathbb{R}^2 \setminus \{(0,0)\}$
QUESTION [9 upvotes]: It is well-known that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus \{(0,0)\}$.
I have two questions.
a) Does there exist a continuous bijection $f: \mathbb{R}^2 \to \mathbb{R}^2 \setminus \{(0,0)\}$ ?
b) Does there exist a continuous bijection $g: \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{R}^2$ ?
Thank you very much for your answers in advance!

REPLY [5 votes]: No: Such maps are open by invariance of domain. They are therefore homeomorphisms. http://en.wikipedia.org/wiki/Invariance_of_domain