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.

Contents

Category of pointed spaces

Operations on pointed spaces

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 '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 $Omega\; X$.