Cone (topology)

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:

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

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:

\{(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_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.

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 $(X,x_0)$ is a pointed space, there is a related construction, the reduced cone, given by

X\times [0,1] / (X\times \left\{0\right\} \cup\left\{x_0\right\}\times [0,1])

With this definition, the natural inclusion $x\mapsto (x,1)$ becomes a based map, where we take $(x_0,0)$ to be the basepoint of the reduced cone.

Cone functor

The map $X\mapsto CX$ induces a functor $C:\bold{Top}\to\bold {Top}$ on the category of topological spaces Top.

References

Allen Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
Cone at PlanetMath.org.

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