Suspension

Remarks

The two classes $X \times \set{0}$ and $X \times \set{1}$ are the two cone points of $SX$. A continuous map $f \colon X \to Y$ induces a continuous map $Sf \colon SX \to SY$ by $[x, t] \mapsto [f(x), t]$.

Example: let $X = S^0 = \set{-1, 1}$, a two-point discrete space. Then $X \times [0, 1]$ is two segments, and the relation joins their lower ends and their upper ends. The result is homeomorphic to the circle $S^1$. More generally, $S(S^n)$ is homeomorphic to $S^{n+1}$ for every $n \ge 0$.

Non-example: collapsing only $X \times \set{1}$ gives the cone $CX$, not the suspension. For $X = S^1$ the cone is a disc, which has a boundary circle, while $SS^1$ is a sphere.

For a pointed space $(X, x_0)$, the reduced suspension $\Sigma X$ also collapses the segment $\set{x_0} \times [0, 1]$.