# 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.

### Definition 1

The **final topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$** is defined as:

- $\tau = \set{U \subseteq X: \forall i \in I: \map {f_i^{-1}} U \in \tau_i} \subseteq \powerset X$

### Definition 2

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

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

## 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
- Final Topology Contains Codomain Topology iff Mappings are Continuous

- Results about
**the final topology**can be found**here**.