TITLE: When is the quotient space of a second countable space second countable?
QUESTION [10 upvotes]: I am a bit confused about this concept because I have read that the quotient space is second countable if the quotient map is open. However, I thought the definition of a quotient map was a surjective, continuous, open mapping. 
Suppose that $X$ is a second countable topological space, and ~ is an equivalence relation. The canonical mapping $q: X \rightarrow X$/~ is a quotient map, so $X$/~ would also be second countable. 
Where am I going wrong with this idea?

REPLY [10 votes]: Take $\mathbb R$ and identify $\mathbb Z$ to a point. This will give you a quotient space that is a point with countably infinitely many loops coming out of it. A set in the quotient is open if and only if its pullback via the quotient map is open. In particular, any open neighborhood of the central point must contain a point from each one of the loops, for otherwise the pullback will have isolated points and hence not be open. 
The quotient map is not open, as for example $(-1/2,1/2)$ does not map to an open set because the image contains the central point but does not include any points from all but 2 of the loops. The quotient space also does not have a countable basis since it is not even first countable: given any countable collection of open sets containing the central point we may, using a diagonal argument, construct an open set containing the central point that does not contain any of them.