Based topological space
A based topological space or a topological space with basepoint is the data of a topological space and a point in the topological space (termed basepoint). A based topological space means a topological space with a basepoint .
Sometimes, we suppress the basepoint, or simply call it for all spaces involved.
We define a map of based topological spaces to be a continuous map that sends the basepoint of one to the basepoint of the other.
As a category
Based topological spaces form a category, with the objects being based topological spaces and the morphisms being basepoint-preserving continuous maps.
The initial as well as the terminal object in the category of based topological spaces is the one-point space. In other words, for any based topological space, there is a unique map to and a unique map from the one-point space. Further, the composite of these two maps is the identity on the one-point space.
Given maps from a based topological space to and , there is a unique map from to the smash product such that the diagram commutes with the projections from the wedge sum to and . Thus, smash product is the correct product notion in the category of based topological spaces.
Given maps from based topological spaces and to , there is a unique map from to such that the diagram commutes with the natural inclusions from each of and to .