nLab pointed space

Concretely, by a pointed space or based space is typically meant a pointed topological space. See there for more.

Generally, a pointed space, or based space, is a pointed object in a category of spaces of sorts.

That is, a pointed space is a space $X$ together with a point $a$ of $X$, called the basepoint, or base point, of the pointed space. Here ‘point’ in general means global element; this matches the usual meaning of ‘point’ in all of the common categories of spaces, even non-concrete ones like the category of locales.

Many constructions in homotopy theory, such as the fundamental group and other homotopy groups, are defined on pointed topological spaces. One may remove the dependence on a basepoint by passing to the fundamental groupoid.