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In topology, especially algebraic topology, the cone C X {\displaystyle CX} of a topological space X {\displaystyle X} is the space defined as:
where v {\displaystyle v} is a point and p {\displaystyle p} is the projection to that point.
That means, the cone C X {\displaystyle CX} is the result of attaching the cylinder X × {\displaystyle X\times } by its face X × { 0 } {\displaystyle X\times \{0\}} to the point v {\displaystyle v} along the projection p : → v {\displaystyle p:{\bigl }\to v}.
If X {\displaystyle X} is a non-empty compact subspace of Euclidean space, the cone on X {\displaystyle X} is homeomorphic to the union of segments from X {\displaystyle X} to any fixed point v ∉ X {\displaystyle v\not \in X} such that these segments intersect only by v {\displaystyle v} 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.