Definition:Initial Topology

Definition
Let $X$ be a set.

Let $I$ be an indexing set.

For each $i \in I$ let:


 * $\left({Y_i, \vartheta_i}\right)$ be a topological space;


 * $f_i: X \to Y_i$ be a mapping.

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

That is, $\mathcal S$ consists of the set of all the preimages of all the open sets of all the topological spaces.

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