Initial topology

In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set ${\displaystyle X,}$ with respect to a family of functions on ${\displaystyle X,}$ is the coarsest topology on ${\displaystyle X}$ that makes those functions continuous.

The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.

The dual notion is the final topology, which for a given family of functions mapping to a set ${\displaystyle X}$ is the finest topology on ${\displaystyle X}$ that makes those functions continuous.