# 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 space defined as:

$CX=(X\times [0,1])\cup _{p}v\ =\ \varinjlim {\bigl (}(X\times [0,1])\hookleftarrow (X\times \{0\})\xrightarrow {p} v{\bigr )},$ where $v$ is a point (called the vertex of the cone) and $p$ is the projection to that point.

That means, the cone $CX$ is the result of attaching the cylinder $X\times [0,1]$ by its face $X\times \{0\}$ to the point $v$ along the projection $p:{\bigl (}X\times \{0\}{\bigr )}\to v$ .

If $X$ is a non-empty compact subspace of Euclidean space, the cone on $X$ is homeomorphic to the union of segments from $X$ to any fixed point $v\not \in X$ such that these segments intersect only by $v$ 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 $\{p\}\times [0,1]$ .
• 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
$\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=1{\mbox{ and }}z=0\}$ is the curved surface of the solid cone:
$\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=(z-1)^{2}{\mbox{ and }}0\leq z\leq 1\}.$ 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

$h_{t}(x,s)=(x,(1-t)s)$ .

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.

## Cone functor

The map $X\mapsto CX$ induces a functor $C\colon \mathbf {Top} \to \mathbf {Top}$ on the category of topological spaces Top. If $f\colon X\to Y$ is a continuous map, then $Cf\colon CX\to CY$ is defined by

$(Cf)([x,t])=[f(x),t]$ ,

where square brackets denote equivalence classes.

## Reduced cone

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

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