In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint that remains unchanged during subsequent discussion, and is kept track of during all operations. Common examples of naturally pointed spaces include spheres (with the north pole as basepoint), closed intervals [0,1] (with 0 as basepoint), and topological groups (with the identity element as basepoint).

Maps of pointed spaces (based maps) are continuous maps This is usually denoted

Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint.

The pointed set concept is less important; It is anyway the case of a pointed discrete space.

Pointed spaces are often taken as a special case of the relative topology, where the subset is a single point. Thus, much of homotopy theory is usually developed on pointed spaces, and then moved to relative topologies in algebraic topology.

Category of pointed spaces

The class of all pointed spaces forms a category Top with basepoint-preserving continuous maps as morphisms. Another way to think about this category is as the comma category Top is any one point space and Top is the category of topological spaces. (This is also called a coslice categoryTop Top) are morphisms in Top for which the following diagram commutes:

preserves basepoints.

is a zero object in Top, while it is only a terminal object in Top.

There is a forgetful functor Top Top which "forgets" which point is the basepoint. This functor has a left adjoint the disjoint union whose single element is taken to be the basepoint.

Operations on pointed spaces

• A subspace is a topological subspace so that the inclusion map is basepoint preserving.

• One can form the quotient under any equivalence relation under the quotient map.

• One can form the product as the topological productserving as the basepoint.

• The coproduct in the category of pointed spaces is the , which can be thought of as the 'one-point union' of spaces.

• The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such as compactly generated weak Hausdorff ones.

• The reduced suspension is (up to a homeomorphism

• The reduced suspension is a functor from the category of pointed spaces to itself. This functor is left adjoint to its loop space.

See also

References

• mathoverflow discussion on several base points and groupoids