pointed topological space

Definition Suppose $X$ is a non-empty topological space and $x_0$ is anelement of $X$. Then the pair $(X,x_0)$ is called apointed topological space, or a based topological space.

The idea with pointed topological spaces is simply that one fixes a base pointin the space. This is necessary, for instance, when defining the fundamentalgroup of a topological space.