Final topology

In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set with respect to a family of functions from topological spaces into is the finest topology on 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 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 into topological spaces is the coarsest topology on that makes those functions continuous. Given a set and an -indexed family of topological spaces with associated functions the is the finest topology on such that is continuous for each . Explicitly, the final topology may be described as follows: a subset of is open in the final topology (that is, ) if and only if is open in for each . The closed subsets have an analogous characterization: a subset of is closed in the final topology if and only if is closed in for each . The family of functions that induces the final topology on is usually a set of functions. But the same construction can be performed if is a proper class of functions, and the result is still well-defined in Zermelo–Fraenkel set theory. In that case there is always a subfamily of with a set, such that the final topologies on induced by and by coincide. For more on this, see for example the discussion here. As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class of functions.