Cone (topology)


In topology, especially algebraic topology, the cone 'of a topological space' 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 or by .

Definitions

Formally, the cone of X is defined as:
where is a point and is the projection to that point. In other words, it is the result of attaching the cylinder by its face to a point along the projection.
If is a non-empty compact subspace of Euclidean space, the cone on is homeomorphic to the union of segments from to any fixed point such that these segments intersect only in 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: the join of with a single point.''''

Examples

Here we often use a geometric cone. 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,.
  • The cone over two points is a "V" shape with endpoints at and.
  • The cone over a closed interval I of the real line is a filled-in triangle, otherwise known as a 2-simplex.
  • The cone over a polygon P is a pyramid with base P.
  • The cone over a disk is the solid cone of classical geometry.
  • The cone over a circle given by
More general examples:
  • The cone over an n-sphere is homeomorphic to the closed -ball.
  • The cone over an n-ball is also homeomorphic to the closed -ball.
  • The cone over an n-simplex is an -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
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, then the cone 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 will be finer than the set of lines joining X to a point.

Cone functor

The map induces a functor on the category of topological spaces Top. If is a continuous map, then is defined by
where square brackets denote equivalence classes.

Reduced cone

If is a pointed space, there is a related construction, the reduced cone, given by
where we take the basepoint of the reduced cone to be the equivalence class of. With this definition, the natural inclusion becomes a based map. This construction also gives a functor, from the category of pointed spaces to itself.