This question was inspired by Lee's Introduction to Topological Manifold problem 12-22 in chapter 12. Here is the original problem:
Give an Example of a manifold $M$ and a discrete group $\varGamma$ acting continuously and properly on $M$, such that $M/\varGamma$ is not a manifold.
I have constructed an example but in my example the orbit space is a manifold with boundary. So I wonder if there is a manifold (with or without boundary) and a discrete group act continuously and properly on it such that the orbit space is even not a manifold with boundary.
And furthermore, if there is a compact manifold (with or without boundary) and a finite group with discrete topology act (in this case it automatically act properly) continuously on it yielding an orbit space which is not a manifold with boundary.
P.S. A continuous action of topological group $G$ on a topological space $E$ was said to be proper iff the map $H$ from $G\times E$ to $E\times E$ given by $H(g,e)=(g\cdot e,e)$ is a proper map (a proper map was defined to be a map (does not need to be continuous) between topological spaces such that the preimage of each compact subset is compact).