I was wondering if anyone visiting would be up for solving the following interesting little exercise out of Fulton's Algebraic Topology: A First Course.
Let $G$ a group act on a set $Y$. Say that $G$ acts evenly on $Y$ if any point in $Y$ has a nbhd. $V$ such that $g \cdot V = \{g \cdot y : y \in V\}$ (the image of $V$ under the group action) and $h \cdot V$ are disjoint for any distinct elements $g$, $h$ in $G$. Prove that the following actions are even:
(1) $\mathbb{Z}$ acts on $\mathbb{R}$ by translation: $n \cdot r = r+n$.
(2) $\mathbb{Z}$ acts on the right half plane (RHP) in $\mathbb{R^2}$ so that the polar coordinate covering map $p: \{(r, \theta) \in \mathbb{R^2}: r>0\} \rightarrow \mathbb{R^2}\setminus \{(0,0)\}$ given by $p(r,\theta) = (r \cos \theta, r \sin \theta),$ is the quotient of the RHP by this $\mathbb{Z}$-action.
(3) The cyclic group $G = \mu_n$ of $n$th roots of unity in $\mathbb{C}$ acts by multiplication on $S^1$ [regarded as the unit circle in the complex plane].