Cone (topology)
|
|
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 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 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 geometric cone (technically only a half-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.
- 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.