Let \(X\) be a topological space and \(I = [0,1]\). The cone on \(X\) is the quotient space \[ CX\coloneqq(X\times I)/(X\times\ts{0}), \] in which \(X\times\ts{0}\) is collapsed to a point.
If \(X\) is nonempty, then \(CX\) is contractible, and \(x\mapsto[(x, 1)]\) is an embedding of \(X\) into \(CX\).
The cone \(CS^1\) on the circle is homeomorphic to the closed disk \(D^2\), by \([(z, t)]\mapsto tz\) for \(z\in S^1\subseteq\CC\) and \(t\in I\).
See [@Hat02].