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The cone of a topological space $X$ is $CX=(X\times [0,1])/(X\times \{0\})$, with one end collapsed to a vertex. Realization of a simplicial cone gives this construction for the realized base complex.

Triangulate $X$ and adjoin one new vertex joined to every simplex; the resulting simplicial complex realizes the cone $CX$. The cone contracts to its apex, so