Final topology

In general topology and related areas of mathematics, the final topology^[1] (or coinduced,^[2] weak, colimit, or inductive^[3] topology) on a set ${\displaystyle X,}$ with respect to a family of functions from topological spaces into ${\displaystyle X,}$ is the finest topology on ${\displaystyle X}$ that makes all those functions continuous.

The quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusion maps. The final topology is also the topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is coherent with some collection of subspaces if and only if it is the final topology induced by the natural inclusions.

The dual notion is the initial topology, which for a given family of functions from a set ${\displaystyle X}$ into topological spaces is the coarsest topology on ${\displaystyle X}$ that makes those functions continuous.