# Pointed space

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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.

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 taking a based space to its loop space .

## References

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