\begin{definition}[Definition:Initial Topology/Definition 1]
Let $X$ be a set.
Let $I$ be an indexing set.
Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$.
Let $\family {f_i: X \to Y_i}_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$.
Let:
:$\SS = \set {f_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$
where $f_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $f_i$.
The topology $\tau$ on $X$ generated by $\SS$ is called the '''initial topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$'''.
\end{definition}