Based topological space

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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.


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.


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.