Definition:Initial Topology

Definition
Let $X$ be a set.

Let $I$ be an indexing set.

Let $\left\langle {\left({Y_i, \tau_i}\right)} \right\rangle_{i \in I}$ be an $I$-indexed family of topological spaces.

Let $\left\langle {f_i: X \to Y_i} \right\rangle_{i \in I}$ be an $I$-indexed family of mappings.

Let:
 * $\mathcal S = \left\{{f_i^{-1} \left({U}\right): i \in I, \, U \in \tau_i}\right\} \subseteq \mathcal P \left({X}\right)$

where $f_i^{-1} \left({U}\right)$ denotes the preimage of $U$ under $f_i$.

The topology $\tau$ on $X$ generated by $\mathcal S$ is called the initial topology on $X$ with respect to $\left \langle {f_i}\right \rangle_{i \in I}$.

Equivalently, $\tau$ is the coarsest topology on $X$ such that $f_i$ is continuous for all $i \in I$.

Also see

 * Final Topology