Topology carried over by a mapping of a topological space into a set. Let $f:X \rightarrow Y$ be a mapping of a topological space $X$ into a set $Y$. Let $J$ be the topology on $X$. What exactly is "the topology carried over to $Y$ by $f$"?
 A: There is a final topology determined by $f$: here the open sets of $Y$ are the subsets $V \subset Y$ such that $f^{-1}(V)$ is open in $X$. This satisfies a universal property: see Proposition 1.7.1 of Schapira's notes. Assuming you want $f$ to be continuous, this topology is as fine as it could be in the sense that it contains any topology which makes $f$ continuous.
If $f$ is surjective then we've just described the quotient topology determined by $f$. I must admit that I can't think of any good examples in which this is not the case.
A: That topology consists of exactly the subsets $U\subseteq Y$ for which $f^{-1}(U) \in J$, that is $f^{-1}(U)$ is open in $X$.
A: One should also emphasise the universal property: with the final topology on $Y$, if $Z$ is a topological space, then  a function $g:Y \to Z$ is continuous if and only if the composition $gf$ is continuous. Many constructions in general topology are designed to enable the construction of continuous functions. The final topology is also called the identification topology in the case $f$ is surjective. 
Dually, there is an initial topology  for $f:X \to Y$ if now $X$ is a set, and $Y$ is a topological space. 
