I'm a bit new to quotient maps and I'm struggling to understand the following:
If a map $f:X \to X$ takes $A \subseteq X$ to $A \subseteq X$, I can see why it immediately induce a quotient map:
$$ f': X \to X/A$$ $$ x \mapsto [f(x)]$$
but I don't see where it might fail if the condition "$f:X \to X$ takes $A \subseteq X$ to $A \subseteq X$" is not satisfied. I tried thinking of the following example:
$ X = [0,1]$ and $A = \{0,1\}$ (which means $X/A = S^1$)
and $$f: X \to X$$ $$0 \mapsto 0.5$$ $$x\neq 0 \mapsto x$$
but can't see what might be wrong with that (it's onto; not sure about the open sets condition, but if I'm not mistaken, it does hold - so maybe this is not a good counter example).
Any advice would be much appreciated.