Initial topology
= Initial topology
{title2=$\sigma(f_i:i\in I)$}
For a family of <functions> $f_i:X\to Y_i$ into <topological spaces>, the <initial topology> on $X$ is the coarsest <topology> making them all <continuous>. Its <subbasis of a topology>[subbasis] consists of $f_i^{-1}(U)$ for open <subsets> $U\subseteq Y_i$. A map into $X$ is <continuous> exactly when all its composites with the $f_i$ are <continuous>.