Definition 4.1.7 (Initial Topology). Let $X$ be a set, $\bracsn{(Y_j, \topo_i)}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_{i}: X \to Y_{i}$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that:
For each $i \in I$, $f_{i} \in C(X; Y_{i})$.
If $\mathcal{S}$ is a topology on $X$ satisfying $(1)$, then $\mathcal{S}\supset \topo$.
The family
\[\mathcal{B}= \bracs{\bigcap_{j \in J}f_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \topo_j}\]is a base for $\topo$.
The topology $\topo$ is known a the initial/weak topology generated by the maps $\seqi{f}$.
Proof. Let $\topo$ be the topology genereated by sets of the form $\ce = \bracs{f_i^{-1}(U_i)| i \in I, U_i \in \topo_i}$. Let $\topo$ be the topology generated by $\ce$, then
For each $i \in I$, $\topo \supset \bracs{f_i^{-1}(U)|U \in \topo_i}$, so $f_{i} \in C(\topo; Y_{i})$.
If $\mathcal{S}$ is a topology such that $f_{i} \in C(X, \mathcal{S}; Y_{i})$, then $\bracs{f_i^{-1}(U)|U \in \topo_i}\subset \mathcal{S}$. Thus $\ce \subset \mathcal{S}$ and $\mathcal{S}\supset \topo$.
By Definition 4.1.6, $\cb$ is a base for $\topo$.
$\square$