There is a well-known counterexample to the claim that quotient spaces of second countable spaces are second countable. It is $\mathbb{R} / \sim$ where $$ x \sim y \iff x = y \text{ or } x, y \in \mathbb{Z}. $$
I want to know if there is a generalization to this. Specifically, is it true that any quotient space of an uncountable second countable space within which we identify countably many points will fail to be second countable? Follow up question: is there a nice necessary and sufficient condition for quotient spaces of second countable spaces being second countable? Maybe if the quotient map is open? I know that's sufficient but I don't know if it's necessary.
No, you an identify $[0,1]$ to a point and the result is homeomorphic to $\Bbb R$ again, and there we even identify uncountably many points. So it has to be decided on a case by case basis. If $X$ is compact, Hausdorff, second countable and we identify a closed set to a point, then $X{/}\sim$ is again second countable. This even holds for such an $X$ that is non-compact (but $T_2$ and $C_2$) and where we identify a compact set to a point. (In that case the quotient map is perfect). If the quotient map is closed and all classes are compact, we have the same general theorem we can use: that $C_2$ is preserved by perfect maps..