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]. | |||
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−''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]] |
Revision as of 03:24, 9 August 2013
In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space:
of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse one end of the cylinder to a point.
If X sits inside Euclidean space, the cone on X is homeomorphic to the 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 is the solid cone of classical geometry (hence the concept's name).
- The cone over a circle is the curved surface of the solid cone:
- This in turn is homeomorphic to the closed disc.
- In general, the cone over an n-sphere is homeomorphic to the closed (n+1)-ball.
- The cone over an n-simplex is an (n+1)-simplex.
Properties
All cones are path-connected since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the homotopy
- ht(x,s) = (x, (1−t)s).
The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space.
When X is compact and 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 than the set of lines joining X to a point.
Reduced cone
If is a pointed space, there is a related construction, the reduced cone, given by
With this definition, the natural inclusion becomes a based map, where we take to be the basepoint of the reduced cone.
Cone functor
The map induces a functor on the category of topological spaces Top.
See also
References
- Allen Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
- Template:Planetmath reference