Definition:Initial Topology

Let $$X, I$$ be sets.

For each $$i \in I$$ let:
 * $$Y_i$$ be a topological space;
 * $$f_i : X \to Y_i$$ a mapping.

Let $$\mathcal{S} := \left\{{f_i^{-1} \left({U}\right) : i \in I, U \subseteq Y_i \text{ open}}\right\}$$.

The generated topology for $$\mathcal{S}$$ on $$X$$ is called the initial topology on $$X$$ with respect to the $$(f_i)_{i \in I}$$.