View source for Pointed space

{{Short description|Topological space with a distinguished point}}

In [[Mathematics|mathematics]], a '''pointed space''' or '''based space''' is a [[Topological space|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 <math>x_0,</math> that remains unchanged during subsequent discussion, and is kept track of during all operations.

Maps of pointed spaces ('''based maps''') are continuous maps preserving basepoints, i.e., a map <math>f</math> between a pointed space <math>X</math> with basepoint <math>x_0</math> and a pointed space <math>Y</math> with basepoint <math>y_0</math> is a based map if it is continuous with respect to the topologies of <math>X</math> and <math>Y</math> and if <math>f\left(x_0\right) = y_0.</math> This is usually denoted
:<math>f : \left(X, x_0\right) \to \left(Y, y_0\right).</math>
Pointed spaces are important in [[Algebraic topology|algebraic topology]], particularly in [[Homotopy theory|homotopy theory]], where many constructions, such as the [[Fundamental group|fundamental group]], depend on a choice of basepoint.

The [[Pointed set|pointed set]] concept is less important; it is anyway the case of a pointed [[Discrete space|discrete space]].

Pointed spaces are often taken as a special case of the relative topology, where the subset is a single point.  Thus, much of [[Homotopy theory|homotopy theory]] is usually developed on pointed spaces, and then moved to relative topologies in [[Algebraic topology|algebraic topology]].

== Category of pointed spaces ==

The [[Class (set theory)|class]] of all pointed spaces forms a [[Category (mathematics)|category]] '''Top'''<sub><math>\bull</math></sub> with basepoint preserving continuous maps as [[Morphism|morphism]]s. Another way to think about this category is as the comma category, (<math>\{ \bull \} \downarrow </math> '''Top''') where <math>\{ \bull \}</math> is any one point space and '''Top''' is the [[Category of topological spaces|category of topological spaces]]. (This is also called a coslice category denoted <math>\{ \bull \} /</math>'''Top'''.) Objects in this category are continuous maps <math>\{ \bull \} \to X.</math> Such maps can be thought of as picking out a basepoint in <math>X.</math> Morphisms in (<math>\{ \bull \} \downarrow </math> '''Top''') are morphisms in '''Top''' for which the following diagram [[Commutative diagram|commutes]]:

<div style="text-align: center;">
[[Image:PointedSpace-01.png]]
</div>

It is easy to see that commutativity of the diagram is equivalent to the condition that <math>f</math> preserves basepoints.

As a pointed space, <math>\{ \bull \}</math> is a zero object in '''Top'''<sub><math>\{ \bull \}</math></sub>, while it is only a terminal object in '''Top'''.

There is a [[Forgetful functor|forgetful functor]] '''Top'''<sub><math>\{ \bull \}</math></sub> <math>\to</math> '''Top''' which "forgets" which point is the basepoint. This functor has a left adjoint which assigns to each topological space <math>X</math> the [[Disjoint union|disjoint union]] of <math>X</math> and a one-point space <math>\{ \bull \}</math> whose single element is taken to be the basepoint.

== Operations on pointed spaces ==

* A '''subspace''' of a pointed space <math>X</math> is a topological subspace <math>A \subseteq X</math> which shares its basepoint with <math>X</math> so that the [[Inclusion map|inclusion map]] is basepoint preserving.
* One can form the '''[[Quotient space (topology)|quotient]]''' of a pointed space <math>X</math> under any [[Equivalence relation|equivalence relation]]. The basepoint of the quotient is the image of the basepoint in <math>X</math> under the quotient map.
* One can form the '''[[Product (category theory)|product]]''' of two pointed spaces <math>\left(X, x_0\right),</math> <math>\left(Y, y_0\right)</math> as the topological product <math>X \times Y</math> with <math>\left(x_0, y_0\right)</math>serving as the basepoint.
* The '''[[Coproduct|coproduct]]''' in the category of pointed spaces is the {{em|[[Wedge sum|wedge sum]]}}, which can be thought of as the 'one-point union' of spaces.
* The '''[[Smash product|smash product]]''' of two pointed spaces is essentially the [[Quotient space (topology)|quotient]] of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 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 space|compactly generated]] [[Weak Hausdorff space|weak Hausdorff]] ones.
* The '''reduced suspension''' <math>\Sigma X</math> of a pointed space <math>X</math> is (up to a [[Homeomorphism|homeomorphism]]) the smash product of <math>X</math> and the pointed circle <math>S^1.</math>
* The reduced suspension is a functor from the category of pointed spaces to itself. This functor is left adjoint to the functor <math>\Omega</math> taking a pointed space <math>X</math> to its [[Loop space|loop space]] <math>\Omega X</math>.

== See also ==

* {{annotated link|Category of groups}}
* {{annotated link|Category of metric spaces}}
* {{annotated link|Category of sets}}
* {{annotated link|Category of topological spaces}}
* {{annotated link|Category of topological vector spaces}}

== References ==

* {{Cite book
|last1=Gamelin
|first1=Theodore W.
|last2=Greene
|first2=Robert Everist
|title=Introduction to Topology
|edition=second
|year=1999
|orig-year=1983
|publisher=[[Social:Dover Publications|Dover Publications]]
|isbn=0-486-40680-6
|url-access=registration
|url=https://archive.org/details/introductiontoto00game
}}
* {{Cite book
|first=Saunders
|last=Mac Lane
|title=[[Categories for the Working Mathematician]]
|edition=second
|date=September 1998
|publisher=Springer
|isbn=0-387-98403-8}}

* [https://mathoverflow.net/q/40945 mathoverflow discussion on several base points and groupoids ]

{{DEFAULTSORT:Pointed Space}}
[[Category:Topology]]
[[Category:Homotopy theory]]
[[Category:Topological spaces]]

{{Sourceattribution|Pointed space}}