Cone (topology)
In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space:
of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse one end of the cylinder to a point.
If X sits inside Euclidean space, the cone on X is homeomorphic to the union of lines from X to another point. That is, the topological cone agrees with the geometric cone when defined. However, the topological cone construction is more general.
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 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.
If is a pointed space, there is a related construction, the reduced cone, given by
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