A pointed space, or based space, is a pointed object in a category of spaces. That is, a pointed space is a space together with a point of , 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 spaces. One can remove the dependence on a basepoint by passing to the fundamental groupoid.