Pointed topological 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 : XY 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.

As a pointed space {•} is a zero object in Top while it is only a terminal object in Top.

There is a forgetful functor TopTop 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 AX 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.

References

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