General Topology/Pointed spaces and support

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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 .

Definition (support):

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.