Cone (topology)

is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.

Formally, the cone of X is defined as: where

is a point (called the vertex of the cone) and

In other words, it is the result of attaching the cylinder

is a non-empty compact subspace of Euclidean space, the cone on

is homeomorphic to the union of segments from

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:

is a non-empty compact subspace of Euclidean space).

The considered spaces are compact, so we get the same result up to homeomorphism.

More general examples:[1]: 77, Exercise.1 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 (essentially, when X can be embedded in Euclidean space), 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.

on the category of topological spaces Top.

is defined by where square brackets denote equivalence classes.

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

This construction also gives a functor, from the category of pointed spaces to itself.