In topology, a branch of mathematics, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points. One views X as "suspended" between these end points. The suspension of X is denoted by SX or susp(X).

There is a variant of the suspension for a pointed space, which is called the reduced suspension and denoted by ΣX. The "usual" suspension SX is sometimes called the unreduced suspension, unbased suspension, or free suspension of X, to distinguish it from ΣX.

The (free) suspension ${\displaystyle SX}$ of a topological space ${\displaystyle X}$ can be defined in several ways.

1. ${\displaystyle SX}$ is the quotient space ${\displaystyle (X\times [0,1])/(X\times \{0\}){\big /}(X\times \{1\}).}$ In other words, it can be constructed as follows:

2. Another way to write this is:

${\displaystyle SX:=v_{0}\cup _{p_{0}}(X\times [0,1])\cup _{p_{1}}v_{1}\ =\ \varinjlim _{i\in \{0,1\}}{\bigl (}(X\times [0,1])\hookleftarrow (X\times \{i\})\xrightarrow {p_{i}} v_{i}{\bigr )},}$

Where ${\displaystyle v_{0},v_{1}}$ are two points, and for each i in {0,1}, ${\displaystyle p_{i}}$ is the projection to the point ${\displaystyle v_{i}}$ (a function that maps everything to ${\displaystyle v_{i}}$). That means, the suspension ${\displaystyle SX}$ is the result of constructing the cylinder ${\displaystyle X\times [0,1]}$, and then attaching it by its faces, ${\displaystyle X\times \{0\}}$ and ${\displaystyle X\times \{1\}}$, to the points ${\displaystyle v_{0},v_{1}}$ along the projections ${\displaystyle p_{i}:{\bigl (}X\times \{i\}{\bigr )}\to v_{i}}$.

3. One can view ${\displaystyle SX}$ as two cones on X, glued together at their base.