---
tag: 2VD7GR64JJ
kind: definition
name: Suspension
msc: [55P40, 54B15]
defines: Q7RFNJH76Q
uses: [0CQHFPGJ45, 0GBG8B49AA, 0G8VEZ3WTK, 0NQ1H2X495, 0QT00PB5G2]
sources:
  - url: https://en.wikipedia.org/wiki/Suspension_(topology)
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], and give $X \times [0, 1]$ the [[product topology|0GBG8B49AA]].
Let $\sim$ be the [[equivalence relation|0G8VEZ3WTK]] on $X \times [0, 1]$ generated by the following relations, for all $x, y \in X$:

1. $(x, 0) \sim (y, 0)$;
2. $(x, 1) \sim (y, 1)$.

The *suspension* of $X$ is the quotient
$$SX = (X \times [0, 1]) / {\sim},$$
with the [[quotient topology|0NQ1H2X495]].

## 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|0QT00PB5G2]] 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 $\set{x_0} \times [0, 1]$.