A pointed topological space, written as $(X,x_0)$ , consists of a non-empty topological space $X$ together with an element $x_0\in X$ . The terminology based topological space is also used often.

If $(X,x_0)$ is a pointed space, we call $X$ its underlying topological space and $x_0$ its basepoint.

A morphism from $(X,x_0)$ to $(Y,y_0)$ is a continuous map $f\co X\to Y$ satisfying $f(x_0)=y_0$ . With these morphisms, the pointed topological spaces form a category.

Two pointed topological spaces $(X,x_0)$ and $(Y,y_0)$ are isomorphic in this category if there exists a homeomorphism $f\co X\to Y$ with $f(x_0)=y_0$ .

Every singleton (a pointed topological space of the form $(\{x_0\}, x_0)$ ) is a zero object in this category.

For every pointed topological space $(X,x_0)$ , we can construct the fundamental group $\pi(X,x_0)$ and for every morphism $f\co (X,x_0)\to(Y,y_0)$ we obtain a group homomorphism $\pi(f)\co\pi(X,x_0)\to \pi(Y,y_0)$ . 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 $\pi_i(X,x_0)$ for $i=2,3,\ldots$ that map into the category of abelian groups and the (based) loop space $\Omega(X,x_0)$ that maps into the category of topological spaces.