Pointed space

In mathematics, a pointed space is a topological space "X" with a distinguished basepoint "x"₀ in "X". Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e. a continuous map "f" : "X" → "Y" such that "f"("x"₀) = "y"₀. This is usually denoted:"f" : ("X", "x"₀) → ("Y", "y"₀).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", "x"₀), ("Y", "y"₀) as the topological product "X" × "Y" with ("x"₀, "y"₀) 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 Σ"X" of a pointed space "X" is (up to a homeomorphism) the smash product of "X" and the pointed circle "S"¹.
*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 $Omega\; X$.