Let $S$ be a second countable topological space. Let $S^*$ be a quotient space of $S$ with quotient map $\pi$. If $\pi$ is open, it's easy to show that it transfers a basis of $S$ into a basis of $S^*$. So, in that case $S$ being second countable space implies that $S^*$ is second countable.
Does this hold for general quotient maps?