Cone (topology)
Definitions
Definitions
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Formally, the cone of *X* is defined as:

CX \= (X \\times \[0,1])\\cup\_p v\\ \=\\ \\varinjlim \\bigl( (X \\times \[0,1]) \\hookleftarrow (X\\times \\{0\\}) \\xrightarrow{p} v\\bigr),
where v is a [point](/wiki/Point_%28topology%29 "Point (topology)") (called the vertex of the cone) and p is the [projection](/wiki/Projection_%28mathematics%29 "Projection (mathematics)") to that point. In other words, it is the result of [attaching](/wiki/Adjunction_space "Adjunction space") the [cylinder](/wiki/Cylinder_%28geometry%29 "Cylinder (geometry)") X \\times \[0,1] by its face X\\times\\{0\\} to a point v along the projection p: \\bigl( X\\times\\{0\\} \\bigr)\\to v. 

If X is a non\-empty [compact](/wiki/Compact_space "Compact space") subspace of [Euclidean space](/wiki/Euclidean_space "Euclidean space"), the cone on X is [homeomorphic](/wiki/Homeomorphic "Homeomorphic") to the [union](/wiki/Union_%28set_theory%29 "Union (set theory)") of segments from X to any fixed point v \\not\\in X such that these segments intersect only in v itself. That is, the topological cone agrees with the [geometric cone](/wiki/Geometric_cone "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](/wiki/Join_%28topology%29 "Join (topology)"): CX \\simeq X\\star \\{v\\} \=  the join of X with a single point v\\not\\in X. 


What is the union of segments from X to any fixed point v called?