# Definition:Final Topology

## Definition

Let $X$ be a set.

Let $I$ be an indexing set.

Let $\left\langle {\left({Y_i, \tau_i}\right)} \right\rangle_{i \mathop \in I}$ 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 $f \left({\tau_Y}\right)$.