# Universal Constructions

Definition (Initial Topology)

Suppose that $f_{i}:S→X_{i}$, $i∈I$, is a family of maps, from a set $S$ into topological spaces $X_{i}$. The

initial topologyon $S$ 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 $g:T→S$ is continuous if and only if all compositions $f_{i}∘g:T→X_{i}$ are continuous.