Let \(X\) be a topological space and \(I=[0,1]\). The suspension \(SX\) is the quotient of \(X\times I\) obtained by collapsing \(X\times\ts{0}\) to one point and \(X\times\ts{1}\) to another point [@Hat02]. For a based space \((X,x_0)\), the reduced suspension is \[ \Sigma X\coloneqq\frac{X\times I}{(X\times\ts{0})\cup(X\times\ts{1})\cup(\ts{x_0}\times I)}, \] the quotient of \(SX\) obtained by collapsing the segment \(\ts{x_0}\times I\) to a point [@Hat02].
\(SX\) is the union of two copies of the cone \(CX=(X\times I)/(X\times\ts{0})\) glued along their copies of \(X\times\ts{1}\cong X\).