Building new topological spaces through canonical maps

Suppose we have some topological spaces lying around. How can we build new topological spaces using the old ones? There are four fundamental constructions: subspaces, disjoint unions, products, and quotients. Defining the topologies on each can be done in two ways. One way is through ad hoc definitions. These definitions make some intuitive sense, but look very different from one construction to the next. The other way uses canonical maps. Canonical maps provide a single framework in which all constructions obey the same unifying principle.