Given a topological space $X,$ the suspension of $X,$ often denoted by $SX,$ is defined to be the quotient space $X\times[0,1]/\sim,$ where $(x,0)\sim(y,0)$ and $(x,1)\sim(y,1)$ for any $x,y\in X.$

Given a continuous map $f:X\rightarrow Y,$ there is a map $Sf:SX\rightarrow SY$ defined by $Sf([x,t]):=[f(x),t].$ This makes $S$ into a functor from the category of topological spaces into itself.

Note that $SX$ is homeomorphic to the join $X\star S^{0},$ where $S^{0}$ is a discrete space with two points.

The space $SX$ is sometimes called the unreduced, unbased or free suspension of $X,$ to distinguish it from the reduced suspension described below.

If $(X,x_{0})$ is a based topological space, the reduced suspension of $X,$ often denoted $\Sigma X$ (or $\Sigma_{{x_{0}}}X$ when the basepoint needs to be explicit), is defined to be the quotient space $X\times[0,1]/(X\times\left\{0\right\}\cup X\times\left\{1\right\}\cup\left\{x_{0}\right\}\times[0,1].$ Setting the basepoint of $\Sigma X$ to be the equivalence class of $(x_{0},0),$ the reduced suspension is a functor from the category of based topological spaces into itself.

An important property of this functor is that it is a left adjoint to the functor $\Omega$ taking a (based) space $X$ to its loop space $\Omega X$. In other words, $\mathop{\mathrm{Maps}_{*}}\left(\Sigma X,Y\right)\cong\mathop{\mathrm{Maps}_{*}}\left(X,\Omega Y\right)$ naturally, where $\mathop{\mathrm{Maps}_{*}}\left(X,Y\right)$ stands for continuous maps which preserve basepoints.

The reduced suspension is also known as the based suspension.