For a space X , the suspension SX is the quotient of X × I obtained by collapsing X × {0} to one point and X × {1} to another point.

Reduced suspension of a space. Let X be a CW complex and x 0 ∈ X a 0 cell. Inside the suspension SX we have the line segment {x 0 }× I , and collapsing this to a point yields a space ΣX homotopy equivalent to SX , called the reduced suspension of X.