Universal Constructions

Definition (Initial Topology)

Suppose that $f_i:S\to X_i$, $i\in I$, is a family of maps, from a set $S$ into topological spaces $X_i$. The initial topology on $S$ induced by the family $(f_i)$ is defined to be the weakest topology on $S$ making all maps $f_i$ continuous.

Universal Property of the Initial Topology

The initial topology on $S$ induced by the family $(f_i)$ is the unique topology on $S$ with the property that for any topological space $T$, a mapping $g:T\to S$ is continuous if and only if all compositions $f_i\circ g:T\to X_i$ are continuous.