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