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 variation of the suspension for 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. Free suspension The (free) suspension of a topological space can be defined in several ways. 1. is the quotient space . In other words, it can be constructed as follows: Construct the cylinder . Consider the entire set as a single point ("glue" all its points together). Consider the entire set as a single point ("glue" all its points together). 2. Another way to write this is: Where are two points, and for each i in {0,1}, is the projection to the point (a function that maps everything to ). That means, the suspension is the result of constructing the cylinder , and then attaching it by its faces, and , to the points along the projections . 3. One can view as two cones on X, glued together at their base. 4. can also be defined as the join where is a discrete space with two points. Properties In rough terms, S increases the dimension of a space by one: for example, it takes an n-sphere to an (n + 1)-sphere for n ≥ 0. Given a continuous map there is a continuous map defined by where square brackets denote equivalence classes. This makes into a functor from the category of topological spaces to itself. Reduced suspension If X is a pointed space with basepoint x0, there is a variation of the suspension which is sometimes more useful. The reduced suspension or based suspension ΣX of X is the quotient space: . This is the equivalent to taking SX and collapsing the line (x0 × I) joining the two ends to a single point. The basepoint of the pointed space ΣX is taken to be the equivalence class of (x0, 0). One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S1. For well-behaved spaces, such as