Pointed space

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.

is a based map if it is continuous with respect to the topologies of

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.

The class of all pointed spaces forms a category Top

with basepoint preserving continuous maps as morphisms.

(This is also called a coslice category denoted

Objects in this category are continuous maps

Such maps can be thought of as picking out a basepoint in

It is easy to see that commutativity of the diagram is equivalent to the condition that

, 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 which assigns to each topological space

whose single element is taken to be the basepoint.