If we take a pointed space$(X,x_0)$, then its reduced suspension$\Sigma X$ is obtained by taking the cylinder $I\times X$ and identifying the subspace $\{0,1\}\times X\cup I\times \{x_0\}$ to a point.

(Think of crushing the two ends of the cylinder and the line through the base point to a point.)

Compare the suspension$S X$, where there is no basepoint and only the ends of the cylinder are crushed.

Definition

For a pointed space $(X,x_0)$,

$\Sigma X = (I\times X)/\{0,1\}\times X\cup I\times \{x_0\}$

This can also be thought of as forming $S^1\wedge X$, the smash product of the circle (based at some point) with $X$: