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.

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 $$(Z,z_0)$$ 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.

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 $$X \vee Y$$.