Cone (topology)

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 topology 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 (topology).

If X sits inside Euclidean space, the cone on X is homeomorphic to the union (set theory) 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 triangle.

The cone over a polygon P is a pyramid with base P .

The cone over a circle inspired the name; CS 1 is homeomorphic to the cone (geometry) (technically only a half-cone):

::{(x,y,z) in mathbb R^3 mid x^2 + y^2 = z^2 mbox{ and } 0leq zleq 1}. :This in turn is homeomorphic to the closed disc (mathematics).

In general, the cone over an n -sphere is homeomorphic to the closed ( n +1)-ball (mathematics).

The cone over an n -simplex is an ( n +1)-simplex.

==Properties==

All cones are path-connected space since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the Homotopy

: h t ( x , s ) = ( x , (1− t ) s ).

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

==See also==

*cone *suspension (topology) *mapping cone *join (topology)