# 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 (*Y*_{i})_{i∈I} 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.