We know that a necessary and sufficient condition for a path-connected, locally path-connected space to have a universal cover is that it is semi-locally simply connected.
Now since $\mathbb R^2\setminus\{0\}$ is such a space, it must have a universal cover. However I can't see what the universal cover of $\mathbb R^2\setminus\{0\}$ actually is. Can someone help me?
Thank you.
Best Answer
Hint: Think of the mapping $z\mapsto e^z$ from $\Bbb{C}$ to $\Bbb{C}\setminus\{0\}$. Its derivative is also $e^z$, which is always non-zero. Therefore the mapping is conformal everywhere, i.e. a local homeomorphism.