[[Glossary of categories]]
# Category of pointed topological spaces

The **category of pointed topological spaces** or **base spaces** $\Top_{•} \simeq \bullet \downarrow \Top$ consists of topological spaces with a basepoint preserving continuous maps as its morphisms. #m/def/topology 
This is a [[Subcategory|full subcategory]] of the **category of topological spaces with base sets** $\Top_{\Omega}$
consisting of topological spaces with a subset of base points as objects
and base set preserving continuous maps as its morphisms, #m/def/topology 
i.e. if $f \in \Top_{\Omega} (XA,YB)$ then $f(A) \sube B$.

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