Definition 5.1.8 (Final Topology).label Let $X$ be a set, $\bracsn{(Y_i, \topo_i)}_{i \in I}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_{i}: Y_{i} \to X$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that:
- (1)
For each $i \in I$, $f_{i} \in C(Y_{i}; X)$.
- (U)
If $\mathcal{S}$ is a topology on $X$ satisfying (1), then $\mathcal{S}\subset \topo$.
- (3)
For any topological space $Z$ and $F: X \to Z$, $F \in C(X; Z)$ if and only if $F \circ f_{i} \in C(Y_{i}; X)$ for all $i \in I$.
The topology $\topo$ is the final topology generated by the maps $\seqi{f}$.
Proof. Let
then since for each $i \in I$, $\topo_{i}$ is a topology on $Y_{i}$, $\topo$ is a topology on $X$.
(1): By definition, for any $i \in I$ and $U \in \topo$, $f_{i}^{-1}(U) \in \topo_{i}$, so $f_{i} \in C(Y_{i}; X)$.
(U): For any topology $\mathcal{S}$ satisfying (1) and $U \in \mathcal{S}$, $f_{i}^{-1}(U) \in \mathcal{T}_{i}$, so $\mathcal{S}\subset \mathcal{T}$.
(3): Let $F: X \to Z$ such that $F \circ f_{i} \in C(Y_{i}; X)$ for all $i \in I$, then for any $U \subset Z$ open, $f_{i}^{-1}(F^{-1}(U)) \in \topo_{i}$ for all $i \in I$. Hence $F^{-1}(U) \in \topo$ and $F \in C(X; Z)$.$\square$