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.

Last revised on November 8, 2018 at 08:15:56. See the history of this page for a list of all contributions to it.