Editing Cone (topology) (section)

== Definitions ==
Formally, the cone of ''X'' is defined as:

:<math>CX = (X \times [0,1])\cup_p v\ =\ \varinjlim \bigl( (X \times [0,1]) \hookleftarrow (X\times \{0\}) \xrightarrow{p} v\bigr),</math>

where <math>v</math> is a [[point (topology)|point]] (called the vertex of the cone) and <math>p</math> is the [[Projection (mathematics)|projection]] to that point. In other words, it is the result of [[adjunction space|attaching]] the [[cylinder (geometry)|cylinder]] <math>X \times [0,1]</math> by its face <math>X\times\{0\}</math> to a point <math>v</math> along the projection <math>p: \bigl( X\times\{0\} \bigr)\to v</math>.  

If <math>X</math> is a non-empty [[compact space|compact]] subspace of [[Euclidean space]], the cone on <math>X</math> is [[homeomorphic]] to the [[union (set theory)|union]] of segments from <math>X</math> to any fixed point <math>v \not\in X</math> such that these segments intersect only in <math>v</math> itself. 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 (topology)|join]]: <math>CX \simeq X\star \{v\} = </math> the join of <math>X</math> with a single point <math>v\not\in X</math>.''<ref name=":0" />''{{Rp|page=76}}