Pointed space

In mathematics, a pointed space is a topological space X with a distinguished basepoint x₀ in X. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f(x₀) = y₀. This is usually denoted

f : (X, x₀) → (Y, y₀).

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.

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) where {•} is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted {•}/Top). Objects in this category are continuous maps {•} → X. Such morphisms can be thought of as picking out a basepoint in X. Morphisms in ({•} ↓ Top) are morphisms in Top for which the following diagram commutes:

It is easy to see that commutativity of the diagram is equivalent to the condition that f preserves basepoints.

Note that as a pointed space {•} 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 which assigns to each topological space X the disjoint union of X and a one point space {•} whose single element is taken to be the basepoint.

Operations on pointed spaces

• A subspace of a pointed space X is a topological subspace A ⊆ X which shares its basepoint with X so that the inclusion map is basepoint preserving.
• One can form the quotient of a pointed space X under any equivalence relation. The basepoint of the quotient is the image of the basepoint in X under the quotient map.
• One can form the product of two pointed spaces (X, x₀), (Y, y₀) as the topological product X × Y with (x₀, y₀) serving as the basepoint.
• The coproduct in the category of pointed spaces is the wedge sum, 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. The smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object.
• The reduced suspension ΣX of a pointed space X is (up to a homeomorphism) the smash product of X and the pointed circle S¹.
• The reduced suspension is a functor from the category of pointed spaces to itself. This functor is a left adjoint to the functor $\Omega$ taking a based space $X$ to its loop space $\Omega X$.