Definition:Final Topology

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.

Also known as
The final topology is also known as the inductive topology.

If only a single topological space $\struct {Y, \tau_Y}$ 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 $\map {f_*} {\tau_Y}$ or $\map f {\tau_Y}$.

Also see

 * Equivalence of Definitions of Final Topology
 * Final Topology is Topology
 * Definition:Initial Topology
 * Final Topology Contains Codomain Topology iff Mappings are Continuous