 Initial topology

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 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.

In topology, the concept of an **initial topology** is a way to construct a topology on a set that reflects the structure imposed by a collection of functions (or maps) from that set to other topological spaces. Specifically, it provides a minimal topology that makes certain maps continuous.