Based topological space

Definition

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 $(X,x_0)$ means a topological space $X$ with a basepoint $x_0$.

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.

Zero object

The zero object in this category is: one-point space

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.

Product

The product in this category is: smash product

Given maps from a based topological space $(Z,z_0)$ to $(X,x_0)$ and $(Y,y_0)$, there is a unique map from $(Z,z_0)$ to the smash product $X\wedge Y$ such that the diagram commutes with the projections from the wedge sum to $X$ and $Y$. Thus, smash product is the correct product notion in the category of based topological spaces.

Coproduct

The coproduct in this category is: wedge sum

Given maps from based topological spaces $(X,x_0)$ and $(Y,y_0)$ to $(Z,z_0)$, there is a unique map from $X\vee Y$ to $(Z,z_0)$ such that the diagram commutes with the natural inclusions from each of $X$ and $Y$ to $X\vee Y$.