The unreduced suspension

Given a topological space the suspension of often denoted by 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 This makes into a functor from the category of topological spaces into itself.

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

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

The reduced suspension

is a based topological space, the reduced suspension of

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

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 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.