Initial topology

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.

Given a set X and an indexed family (Yi)iI of topological spaces with functions

Several topological constructions can be regarded as special cases of the initial topology.

Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.

If a space X comes equipped with a topology, it is often useful to know whether or not the topology on X is the initial topology induced by some family of maps on X. This section gives a sufficient (but not necessary) condition.