Definition:Final Topology

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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:

and can also be denoted by $\map {f_*} {\tau_Y}$ or $\map f {\tau_Y}$.


Also see