Definition:Initial Topology/Definition 1

Definition
Let $X$ be a set.

Let $I$ be an indexing set.

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

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

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 $\left \langle {f_i}\right \rangle_{i \mathop \in I}$.

Also see

 * Equivalence of Definitions of Initial Topology