Cone (topology) Guide, Meaning , Facts, Information and Description

In topology, especially algebraic topology, the cone CX of a topological space X is the quotient

of the product of X with the unit interval [0,1]. Intuitively we make X into part of a cylinder and collapse one end to a point.

This is (homeomorphic to) the space of the union (set theory) of lines from X to another point, but is a more general construction.

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 triangle.
• The cone over a polygon P is a pyramid with base P.
• The cone over a circle inspired the name; CS¹ is homeomorphic to the space (which is only technically a half-cone)

All cones are contractible to the vertex point by the homotopy (represented by)

h_t(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.

See also suspension (topology).

For different notions of cone, see cone.

This is an Article on Cone (topology). Page Contains Information, Facts Details or Explanation Guide About Cone (topology)

	Web www.E-paranoids.com