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.