Pointed space

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In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as ${\displaystyle x_{0},}$ that remains unchanged during subsequent discussion, and is kept track of during all operations.

Maps of pointed spaces (based maps) are

preserving basepoints, i.e., a map ${\displaystyle f}$ between a pointed space ${\displaystyle X}$ with basepoint ${\displaystyle x_{0}}$ and a pointed space ${\displaystyle Y}$ with basepoint ${\displaystyle y_{0}}$ is a based map if it is continuous with respect to the topologies of ${\displaystyle X}$ and ${\displaystyle Y}$ and if ${\displaystyle f\left(x_{0}\right)=y_{0}.}$ This is usually denoted

${\displaystyle f:\left(X,x_{0}\right)\to \left(Y,y_{0}\right).}$

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.

Pointed spaces are often taken as a special case of the

.

Category of pointed spaces

The class of all pointed spaces forms a category Top_{${\displaystyle \bullet }$} with basepoint preserving continuous maps as morphisms. Another way to think about this category is as the comma category, (${\displaystyle \{\bullet \}\downarrow }$ Top) where ${\displaystyle \{\bullet \}}$ is any one point space and Top is the

denoted ${\displaystyle \{\bullet \}/}$Top.) Objects in this category are continuous maps ${\displaystyle \{\bullet \}\to X.}$ Such maps can be thought of as picking out a basepoint in ${\displaystyle X.}$ Morphisms in (${\displaystyle \{\bullet \}\downarrow }$ 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 ${\displaystyle f}$ preserves basepoints.

As a pointed space, ${\displaystyle \{\bullet \}}$ is a

in Top_{${\displaystyle \{\bullet \}}$}, while it is only a in Top.

There is a forgetful functor Top_{${\displaystyle \{\bullet \}}$} ${\displaystyle \to }$ Top which "forgets" which point is the basepoint. This functor has a

which assigns to each topological space ${\displaystyle X}$ the disjoint union of ${\displaystyle X}$ and a one-point space ${\displaystyle \{\bullet \}}$ whose single element is taken to be the basepoint.

Operations on pointed spaces

• A subspace of a pointed space ${\displaystyle X}$ is a
  topological subspace
  ${\displaystyle A\subseteq X}$ which shares its basepoint with ${\displaystyle X}$ so that the inclusion map is basepoint preserving.
• One can form the quotient of a pointed space ${\displaystyle X}$ under any equivalence relation. The basepoint of the quotient is the image of the basepoint in ${\displaystyle X}$ under the quotient map.
• One can form the product of two pointed spaces ${\displaystyle \left(X,x_{0}\right),}$ ${\displaystyle \left(Y,y_{0}\right)}$ as the
  topological product
  ${\displaystyle X\times Y}$ with ${\displaystyle \left(x_{0},y_{0}\right)}$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
  0-sphere as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such as compactly generated weak Hausdorff
  ones.
• The
  reduced suspension
  ${\displaystyle \Sigma X}$ of a pointed space ${\displaystyle X}$ is (up to a homeomorphism) the smash product of ${\displaystyle X}$ and the pointed circle ${\displaystyle S^{1}.}$
• The reduced suspension is a functor from the category of pointed spaces to itself. This functor is
  left adjoint
  to the functor ${\displaystyle \Omega }$ taking a pointed space ${\displaystyle X}$ to its loop space ${\displaystyle \Omega X}$.

See also

• Category of groups – category in mathematicsPages displaying wikidata descriptions as a fallback
• Category of metric spaces – mathematical category with metric spaces as its objects and distance-non-increasing maps as its morphismsPages displaying wikidata descriptions as a fallback
• Category of sets – Category in mathematics where the objects are sets
• Category of topological spaces – category whose objects are topological spaces and whose morphisms are continuous mapsPages displaying wikidata descriptions as a fallback
• Category of topological vector spaces – Topological category

References

• Gamelin, Theodore W.; Greene, Robert Everist (1999) [1983]. Introduction to Topology (second ed.).
  ISBN 0-486-40680-6
  .
• ISBN 0-387-98403-8
  .

• mathoverflow discussion on several base points and groupoids