Bohr–Mollerup theorem: Difference between revisions

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[[Image:Cone.svg|right|thumb|250px|Cone of a circle. The original space is in blue, and the collapsed end point is in green.]]
In [[topology]], especially [[algebraic topology]], the '''cone''' ''CX'' '''of a [[topological space]]''' ''X'' is the [[quotient space]]:


:<math>CX = (X \times I)/(X \times \{0\})\,</math>


of the [[product topology|product]] of ''X'' with the [[unit interval]] ''I'' = [0, 1].  
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Intuitively we make ''X'' into a [[cylinder (geometry)|cylinder]] and collapse one end of the cylinder to a [[point (topology)|point]].
 
If ''X'' sits inside [[Euclidean space]], the cone on ''X'' is [[homeomorphic]] to the [[union (set theory)|union]] of lines from ''X'' to another point. That is, the topological cone agrees with the geometric cone when defined. However, the topological cone construction is more general.
 
==Examples==
 
* The cone over a point ''p'' of the real line is the interval {''p''} x [0,1].
* The cone over two points {0,1} is a "V" shape with endpoints at {0} and {1}.
* The cone over an interval ''I'' of the real line is a filled-in [[triangle]], otherwise known as a 2-simplex (see the final example).  
* The cone over a [[polygon]] ''P'' is a pyramid with base ''P''.
* The cone over a [[disk (mathematics)|disk]] is the solid [[cone (geometry)|cone]] of classical geometry (hence the concept's name).
* The cone over a [[circle]] is the curved surface of the solid cone:
::<math>\{(x,y,z) \in \mathbb R^3 \mid x^2 + y^2 = z^2 \mbox{ and } 0\leq z\leq 1\}.</math>
:This in turn is homeomorphic to the closed [[disc (mathematics)|disc]].
* In general, the cone over an [[n-sphere]] is homeomorphic to the closed (''n''+1)-[[ball (mathematics)|ball]].
* The cone over an ''n''-[[simplex]] is an (''n''+1)-simplex.
 
==Properties==
 
All cones are [[path-connected space|path-connected]] since every point can be connected to the vertex point. Furthermore, every cone is [[contractible]] to the vertex point by the [[homotopy]]
 
:''h''<sub>''t''</sub>(''x'',''s'') = (''x'', (1&minus;''t'')''s'').
 
The cone is used in algebraic topology precisely because it embeds a space as a [[subspace (topology)|subspace]] of a contractible space.
 
When ''X'' is [[compact space|compact]] and [[Hausdorff space|Hausdorff]] (essentially, when ''X'' can be embedded in Euclidean space), then the cone ''CX'' can be visualized as the collection of lines joining every point of ''X'' to a single point. However, this picture fails when ''X'' is not compact or not Hausdorff, as generally the [[quotient topology]] on ''CX'' will be [[finer topology|finer]] than the set of lines joining ''X'' to a point.
 
==Reduced cone==
If <math>(X,x_0)</math> is a [[pointed space]], there is a related construction, the '''reduced cone''', given by
:<math>X\times [0,1] / (X\times \left\{0\right\})
\cup(\left\{x_0\right\}\times [0,1])</math>
 
With this definition, the natural inclusion  <math>x\mapsto (x,1)</math> becomes a based map, where we take <math>(x_0,0)</math> to be the basepoint of the reduced cone.
 
==Cone functor==
The map <math>X\mapsto CX</math> induces a [[functor]] <math>C:\bold{Top}\to\bold {Top}</math> on the [[category of topological spaces]] '''Top'''.
 
==See also==
*[[Cone (disambiguation)]]
*[[Suspension (topology)]]
*[[Mapping cone]]
*[[Join (topology)]]
 
==References==
*[[Allen Hatcher]], [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic topology.''] Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
*{{planetmath reference|id=3974|title=Cone}}
 
[[Category:Topology]]
[[Category:Algebraic topology]]

Latest revision as of 18:21, 27 December 2014


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