|
A pointed topological space, written as  , consists of a non-empty topological space  together with an element  . The terminology based topological space is also used often.
If  is a pointed space, we call  its underlying topological space and  its basepoint.
A morphism from  to  is a continuous map
 satisfying
 . With these morphisms, the pointed topological spaces form a category.
Two pointed topological spaces  and  are isomorphic in this category if there exists a homeomorphism
 with
 .
Every singleton (a pointed topological space of the form
 ) is a zero object in this category.
For every pointed topological space  , we can construct the fundamental group
 and for every morphism
 we obtain a group homomorphism
 . This yields a functor from the category of pointed topological spaces to the category of groups.
Other interesting functors defined on the category of pointed spaces include the higher homotopy groups
 for
 that map into the category of abelian groups and the (based) loop space
 that maps into the category of topological spaces.
| 
