TITLE: Every retraction is a quotient map?
QUESTION [6 upvotes]: I have to proof that every retraction is a quotient map.. I have no idea where to start or what to use! A retraction $r:X \rightarrow A$  is a continuous map s.t. $r(a)=a$ for every $a\in A$.

REPLY [2 votes]: If you don't want to play around with open sets, you could use the universal property of quotient maps: a continuous map $q:X\to A$ is a quotient map if and only if, whenever $f:X\to Z$ is a continuous map such that $q(x)=q(y)\Rightarrow f(y)=f(y)$, there is a unique continuous map $f':A\to Z$ such that $f = f'\circ q$. I suggest that in the case where $q$ is a retraction onto $A\subset X$, given such an $f$, there is a clear choice of $f'$ to try.