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| In [[mathematics]], a '''pointed space''' is a [[topological space]] ''X'' with a distinguished '''basepoint''' ''x''<sub>0</sub> in ''X''. Maps of pointed spaces ('''based maps''') are [[continuous (topology)|continuous maps]] preserving basepoints, i.e. a continuous map ''f'' : ''X'' → ''Y'' such that ''f''(''x''<sub>0</sub>) = ''y''<sub>0</sub>. This is usually denoted
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| :''f'' : (''X'', ''x''<sub>0</sub>) → (''Y'', ''y''<sub>0</sub>).
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| 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.
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| The [[pointed set]] concept is less important; it is anyway the case of a pointed [[discrete space]].
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| ==Category of pointed spaces==
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| The [[class (set theory)|class]] of all pointed spaces forms a [[category (mathematics)|category]] '''Top'''<sub>•</sub> with basepoint preserving continuous maps as [[morphism]]s. 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 [[commutative diagram|commutes]]:
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| <div style="text-align: center;">
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| [[Image:PointedSpace-01.png]]
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| </div>
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| It is easy to see that commutativity of the diagram is equivalent to the condition that ''f'' preserves basepoints.
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| As a pointed space {•} is a [[zero object]] in '''Top'''<sub>•</sub> while it is only a [[terminal object]] in '''Top'''.
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| There is a [[forgetful functor]] '''Top'''<sub>•</sub> → '''Top''' which "forgets" which point is the basepoint. This functor has a [[adjoint functor|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.
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| ==Operations on pointed spaces==
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| *A '''subspace''' of a pointed space ''X'' is a [[subspace (topology)|topological subspace]] ''A'' ⊆ ''X'' which shares its basepoint with ''X'' so that the [[inclusion map]] is basepoint preserving.
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| *One can form the '''[[quotient space|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.
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| *One can form the '''[[product (category theory)|product]]''' of two pointed spaces (''X'', ''x''<sub>0</sub>), (''Y'', ''y''<sub>0</sub>) as the [[product (topology)|topological product]] ''X'' × ''Y'' with (''x''<sub>0</sub>, ''y''<sub>0</sub>) serving as the basepoint.
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| *The '''[[coproduct]]''' in the category of pointed spaces is the ''[[wedge sum]]'', which can be thought of as the one-point union of spaces.
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| *The '''[[smash product]]''' of two pointed spaces is essentially the [[quotient space|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.
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| *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''<sup>1</sup>.
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| *The reduced suspension is a functor from the category of pointed spaces to itself. This functor is a [[left adjoint]] to the functor <math>\Omega</math> taking a based space <math>X</math> to its [[loop space]] <math>\Omega X</math>.
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| ==References==
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| * {{Cite book
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| |last1=Gamelin
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| |first1=Theodore W.
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| |last2=Greene
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| |first2=Robert Everist
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| |title=Introduction to Topology
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| |edition=second
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| |year=1999
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| |origyear=1983
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| |publisher=[[Dover Publications]]
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| |isbn=0-486-40680-6
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| }}
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| * {{Cite book
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| |first=Saunders
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| |last=Mac Lane
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| |authorlink=Saunders Mac Lane
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| |title=[[Categories for the Working Mathematician]]
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| |edition=second
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| |date=September 1998
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| |publisher=Springer
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| |isbn=0-387-98403-8}}
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| {{DEFAULTSORT:Pointed Space}}
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| [[Category:Topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Category-theoretic categories]]
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| [[Category:Topological spaces]]
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