I am reading a book which says that whenever we can define a topology by saying "its the largest topology which satisfy $p$" then it is possible to define the same topology by saying it is the "smallest topology which satisfy $q$". Why is that? Here is an example:
Consider $f:X\rightarrow Y$ and a given topology on $X$, then there is a largest (or finest) topology on $Y$ which makes $f$ continuous. But, the very same topology on $Y$ can be defined as the smallest (or coarsest) topology on $Y$ which satisfy the property: for every other topological space $Z$ and $g:Y\rightarrow Z$, the continuity of $g\circ f$ implies the continuity of $g$.
I need more elaboration on this or a scratch of a proof. (Even though, I assume it is obvious for most people). Many thanks.