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?"



Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Hahn–Mazurkiewicz Theorem: Suppose $X$ is a nonempty Hausdorff topological space.
Then the following are equivalent:


1. there is a surjection $[0,1]\to X$,
2. $X$ is compact, connected, locally connected and second-countable.


It follows that a Hausdorff space satisfying the conditions of (2) is a quotient
of $I = [0,1]$.


Cor: Every connected compact manifold is a quotient of $I$.


Since $I$ is a quotient of $\mathbb{R}^n$, we have your answer.


Cor: Every compact connected $m$-manifold is a quotient of $\mathbb{R}^n$ for any $n\geq 1$.


