Definition:Final Topology/Definition 2

Definition
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.

Let $\tau$ be the finest topology on $X$ such that each $f_i: X \to Y_i$ is $\tuple{\tau, \tau_i}$-continuous.

Then $\tau$ is known as the final topology on $X$ with respect to $\family{f_i}_{i \mathop \in I}$.

Also see

 * Equivalence of Deinitions of Final Topology