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. 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. See also Cone (disambiguation) Suspension (topology) Desuspension Mapping cone (topology) Join (topology) References Allen Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. and Topology Algebraic topology