\begin{definition}[Definition:Final 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 $I$-indexed family of topological spaces.
Let $\family {f_i: Y_i \to X}_{i \mathop \in I}$ be an $I$-indexed family of mappings.
The '''final topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$''' is defined as:
:$\tau = \set{U \subseteq X: \forall i \in I: \map {f_i^{-1}} U \in \tau_i} \subseteq \powerset X$
\end{definition}