Definition:Final Topology

Definition
Let $X$ be a set.

Let $I$ be an indexing set.

Let $\left\langle {\left({Y_i, \tau_i}\right)} \right\rangle_{i \in I}$ be an $I$-indexed family of topological spaces.

Let $\left\langle {f_i: Y_i \to X} \right\rangle_{i \in I}$ be an $I$-indexed family of mappings.

The final topology on $X$ with respect to $\left\langle {f_i} \right\rangle_{i \in I}$ is defined as:
 * $\tau = \left\{{U \subseteq X: \forall i \in I: f_i^{-1} \left({U}\right) \in \tau_i}\right\} \subseteq \mathcal P \left({X}\right)$

Equivalently, $\tau$ is the finest topology on $X$ such that $f_i$ is continuous for all $i \in I$.

Also see

 * Final Topology is Topology
 * Initial Topology