# 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:

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

of the product of X with the unit interval I = [0, 1]. Intuitively, this construction makes X into a cylinder and collapses one end of the cylinder to a point.

If ${\displaystyle X}$ is a compact subspace of Euclidean space, the cone on ${\displaystyle X}$ is homeomorphic to the union of segments from ${\displaystyle X}$ to any fixed point ${\displaystyle v\not \in X}$ such that these segments intersect only by ${\displaystyle v}$ itself. That is, the topological cone agrees with the geometric cone for compact spaces when the latter is defined. However, the topological cone construction is more general.

## Examples

Here we often use geometric cone (defined in the introduction) instead of the topological one. The considered spaces are compact, so we get the same result up to homeomorphism.

• 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 a closed interval I of the real line is a filled-in triangle (with one of the edges being I), 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 given by
${\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=1{\mbox{ and }}z=0\}}$

is the curved surface of the solid cone:

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

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.

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.

## Cone functor

The map ${\displaystyle X\mapsto CX}$ induces a functor ${\displaystyle C:\mathbf {Top} \to \mathbf {Top} }$ on the category of topological spaces Top. If ${\displaystyle f:X\to Y}$is a continuous map, then ${\displaystyle Cf:CX\to CY}$is defined by ${\displaystyle (Cf)([x,t])=[f(x),t]}$, where square brackets denote equivalence classes.

## Reduced cone

If ${\displaystyle (X,x_{0})}$ is a pointed space, there is a related construction, the reduced cone, given by

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

where we take the basepoint of the reduced cone to be the equivalence class of ${\displaystyle (x_{0},0)}$. With this definition, the natural inclusion ${\displaystyle x\mapsto (x,1)}$ becomes a based map. This construction also gives a functor, from the category of pointed spaces to itself.