I'm curious about the following: Is every real $n$-manifold isomorphic to a quotient of $\mathbb{R}^n$?

Thanks.

EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking about topological quotients. Specifically, is there a surjective map $\mathbb{R}^n\to M$ such that $M$ has the quotient topology?

EDIT': I guess an interesting addendum to the question is "when is it true?"