Cone (topology)

In the mathematical branch of topology, a cone over a space is constructed from this point a certain amount of re- forms in a natural way even a topological space. In the Euclidean case, this is actually homeomorphic to a geometric cone, but in general, the topological definition is more inclusive. Mainly cones are considered on topological spaces in algebraic topology.

Definition

Be a topological space. Of the cone is defined as the amount of about

Equipped with the quotient topology with respect to the canonical projection.

The term originates from the Latin word for cone conus.

In detail this means:

Let be a topological space and the real unit interval with the subspace topology. Be on through to the product of these two spaces

An equivalence relation explains. Set now

As the factor space and consider the canonical projection

A subset is now just hot then open when a model, is open in. The system of open sets in effect forms a topology, so the resulting space is the cone over.

Figuratively speaking, the top surface of the cylinder is beaten to a single point.

Properties

• Every topological space can be regarded as part of its space cone with by identifying.

• The cone of a non- empty space is always contractible, by virtue of the homotopy. Together with the first property there is a canonical embedding of an arbitrary ( non-empty ) in a contractible space, which establishes the importance of the cone in algebraic topology.

• The cone over a topological space is homeomorphic to the mapping cone of the identity of this space.

• Each cone is path connected, and notably in the contiguous.

• Folds into a Euclidean space embedding, so is homeomorphic to a geometric cone. Of special importance is the case here, that a coherent, compact subset of. (see examples)

• More generally compact and Hausdorff, his cone of the association corresponds to all routes of points to a common point.

• CW is a complex, as well.

Examples

• The cone over a simplex is a simplex. For a point in particular is a route, a triangle and a tetrahedron.

• In general the homeomorphism.

Reduced cone

Let now a dotted space, so the reduced cone is defined as

With the quotient topology.

With as a base point, even back to a dot area and the above -mentioned inclusion of a base point preserving embedding.

The reduced cone is equal to the reduced mapping cone of the identity.

Kegelfunktor

In category theory, the assignment induces a Endofunktor on the category of topological spaces.

This also assigns to each continuous map to indicate the figure, which is explained by.

The same applies in the category of topological spaces dotted.

Note: The notation used here should not be confused with the mapping cone for constant or the function space of all continuous maps on a topological space.