initial topology

Let $X_i$, $i\in I$ be any family of topological spaces.We say that a topology $𝒯$ on $X$ is initial with respect to the family of mappings $f_i:X\to X_i$, $i\in I$, if $𝒯$ is the coarsest topology on $X$ which makes all $f_i$’s continuous.

The initial topology is characterized by the condition that a map $g:Y\to X$ is continuous if and only if every $f_i\circ g:Y\to X_i$ is continuous.

Sets $𝒮=\{f_i^{-1}(U):U$ is open in $X_i\}$ form a subbase for the initial topology, their finite intersections form a base.

E.g. the product topology is initial with respect to the projections (http://planetmath.org/GeneralizedCartesianProduct) and a subspace topology is initial with respect to the embedding.

The initial topology is sometimes called topology generated by a family of mappings [2], weak topology [4] or projective topology. (The weak topology is used mainly in functional analysis.)

From the viewpoint of category theory, the initial topology is an initial source. (Initial structures, which are a natural generalization of the initial topology, play an important rôle in topological categories and categorical topology.)