Let \(X\) be a second countable space and \(M\) a quotient space of \(X\). If \(M\) is also locally Euclidean it is second countable.
Proof
The space \(M\) can be covered by coordinate balls
, since it is locally euclidean. Then the preimages of that cover covers \(P\) and since it is second countable, we find a countable subcover. Transforming back leads to a countable cover with coordinate balls. Since they are second countable \(M\) is a countable union of second countable spaces and therefore also second countable.
Remark
If \(M\) is also Hausdorff it is a manifold.