Definition:Final Topology/Definition 1

Definition
Let $X$ be a set.

Let $I$ be an indexing set.

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

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

The final topology on $X$ with respect to $\left\langle {f_i} \right\rangle_{i \mathop \in I}$ 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 known as
The final topology is also known as the inductive topology.

If only a single topological space $\left({Y, \tau_Y}\right)$ and a single mapping $f: Y \to X$ are under consideration, the final topology on $X$ with respect to $f$ is additionally known as the:
 * pushforward topology on $X$ under $f$
 * topology on $X$ co-induced by $f$
 * direct image of $\tau_Y$ under $f$
 * identification topology

and can also be denoted by $f_* \left({\tau_Y}\right)$ or $f \left({\tau_Y}\right)$.

Also see

 * Final Topology is Topology
 * Definition:Initial Topology