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In Topology , especially Algebraic Topology , the cone ''CX'' '''of a Topological Space ''' ''X'' is the Quotient Space :

:CX = (X imes I)/(X imes \{0\})\,

of the Product of ''X'' with the Unit Interval ''I'' = 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 {Link without Title} .

  • 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:

    • ::\{(x,y,z) \in \mathbb R^3 \mid x^2 + y^2 = z^2 \mbox{ and } 0\leq z\leq 1\}.

: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

h


The cone is used in algebraic topology precisely because it embeds a space as a Subspace of a contractible space.


SEE ALSO



REFERENCES