Cone (topology)

In topology, especially algebraic topology, the cone of a topological space $$X$$ is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by $$CX$$ or by $$\operatorname{cone}(X)$$.

Definitions
Formally, the cone of X is 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. In other words, it is the result of attaching the cylinder $$X \times [0,1]$$ by its face $$X\times\{0\}$$ to a 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 in $$v$$ 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.

The cone is a special case of a join: $$CX \simeq X\star \{v\} = $$ the join of $$X$$ with a single point $$v\not\in X$$. 

Examples
Here we often use a geometric cone ($$C X$$ where $$X$$ is a non-empty compact subspace of Euclidean space). 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 a line-segment in $$\mathbb{R}^2$$, $$\{p\} \times [0,1]$$.
 * The cone over two points {0,&thinsp;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
 * $$\{(x,y,z) \in \R^3 \mid x^2 + y^2 = 1 \mbox{ and } z=0\}$$
 * is the curved surface of the solid cone:
 * $$\{(x,y,z) \in \R^3 \mid x^2 + y^2 = (z-1)^2 \mbox{ and } 0\leq z\leq 1\}.$$
 * This in turn is homeomorphic to the closed disc.

More general examples: 
 * The cone over an n-sphere is homeomorphic to the closed (n +&thinsp;1)-ball.
 * The cone over an n-ball is also homeomorphic to the closed (n +&thinsp;1)-ball.
 * The cone over an n-simplex is an (n +&thinsp;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.