where last factor is not simply connected: Geometrically, it is seen (for visualization purposes, formula_89 may be replaced by formula_90) that a path from to in formula_91 "is" a loop in formula_92 since and are antipodal points, and that it is not contractible to a point. But a path from to , thence to again, a loop in formula_89 and a "double loop" (considering , where formula_94 is the covering map) in formula_92 that "is" contractible to a point (continuously move away from "upstairs" in formula_89 and shrink the path there to the point ). Thus is a group with two equivalence classes of loops as its elements, or put more simply, is "doubly connected".