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: of the Product of ''X'' with the Unit Interval ''I'' = 1 . Intuitively we make ''X'' into a Cylinder and collapse both ends to two points. One views ''X'' as "suspended" between the end points. One can also view the suspension as two Cones on ''X'' Glued Together at their base (or as a quotient of a single cone). Suspension gives rise to a to an (''n'' + 1)-sphere for ''n'' ≥ 0. The suspension can be used to construct a homomorphism of Homotopy Group s, to which the Freudenthal Suspension Theorem applies. In Homotopy Theory , the phenomena which are preserved under suspension, in a suitable sense, make up Stable Homotopy Theory . REDUCED SUSPENSION If ''X'' is a Pointed Space (with basepoint ''x''0) there is a variation of the suspension with is sometimes more useful. The reduced suspension Σ''X'' of ''X'' is the quotient space: :. This is the equivalent to taking ''SX'' and collapsing the line (''x''0 × ''I'') joining the two ends to a single point. The basepoint of Σ''X'' is the Equivalence Class of (''x''0, 0). Σ then gives rise to a functor from the Category Of Pointed Spaces to itself. One can show that the reduced suspension of ''X'' is Homeomorphic to the Smash Product of ''X'' with the Unit Circle ''S''1. : For Well-behaved spaces, such as CW Complex es, the reduced suspension of ''X'' is Homotopy Equivalent to the ordinary suspension. SEE ALSO REFERENCES
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