\begin{definition}[Definition:Identification Topology]
Let $\struct {S_1, \tau_1}$ be a topological space.
Let $S_2$ be a set.
Let $f: S_1 \to S_2$ be a mapping.
The '''identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$''' is defined as:
:$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$
That is, it is the set of all subsets of $S_2$ whose preimages under $f$ are elements of $\tau_1$.
The '''identification topology''' is seen to depend both on $f$ and $\tau_1$.
\end{definition}