Let $(Y, \tau)$ be a topological space and let $X$ be a set such that there exist a surjective function $f \colon X \to Y$.
Consider $\tau_1$ the smallest topology in $X$ that makes $f$ a quotient map,
$\tau_2$ the smallest topology $X$ that makes $f$ continuous,
$\tau_3$ the smallest topology in $X$ that makes $f$ an open map,
$\tau_4$ the smallest topology in $X$ that makes $f$ a closed map,
$\tau_5$ the smallest topology in $X$ that makes $f$ an open and closed map,
$\tau_6$ the smallest topology in $X$ that makes $f$ a closed and continuous map.
I am asked to compare each topology, I understand the definition for "the smallest topology" but it still troubles me, because I don't know how to use it. Can someone give me an example of comparing these topologies?