Cone (Topology)

In topology, the cone on a space X is constructed by taking the cylinder formed by X and the interval from zero to one, and then collapsing one entire end of that cylinder, the copy of X at zero, down to a single point called the vertex. Every cone is path-connected, since any point in it can be joined to the vertex by a path running along the cylinder, and every cone is contractible to its vertex, meaning it can be continuously shrunk down to that one point. When X is a compact subspace of Euclidean space, this topological construction matches the ordinary geometric cone: the set of all line segments joining points of X to the vertex. Because a cone embeds any space as a subspace of a space that is contractible, it is a standard tool in algebraic topology for studying the properties of the original space, and it can also be described as a special case of the join of two spaces, joining X with a single isolated point.