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?