Definition (pointed topological space):
A pointed topological space is a pair where is a topological space and .
Definition (morphism of pointed topological spaces):
A morphism of pointed topological spaces , and is a continuous function such that .
Let be a continuous map, where is a distinguished point. Then the support of is defined to be
Often, is a topological magma with identity, and is the identity. For instance, and would be a possibility.