0 ) ▁is ▁not . ▁If ▁n ▁> ▁ 2 , ▁then ▁both ▁R n ▁and ▁R n ▁minus ▁the ▁origin ▁are ▁simply ▁connected . ▁ ▁Anal og ously : ▁the ▁n - dimensional ▁sphere ▁Sn ▁is ▁simply ▁connected ▁if ▁and ▁only ▁if ▁n ▁ ≥ ▁ 2 . ▁ ▁Every ▁convex ▁subset ▁of ▁R n ▁is ▁simply ▁connected . ▁ ▁A ▁tor us , ▁the ▁( elli ptic ) ▁cyl inder , ▁the ▁M ö b ius ▁strip , ▁the ▁project ive ▁plane ▁and ▁the ▁Klein ▁bott le ▁are ▁not ▁simply ▁connected . ▁ ▁Every ▁topological ▁vector ▁space ▁is ▁simply ▁connected ; ▁this ▁includes ▁Ban ach ▁spaces ▁and ▁Hil bert ▁spaces . ▁ ▁For ▁n ▁ ≥ ▁ 2 , ▁the ▁special ▁orthogonal ▁group ▁SO ( n , R ) ▁is ▁not ▁simply ▁connected ▁and ▁the ▁special ▁unit ary ▁group ▁SU ( n ) ▁is ▁simply ▁connected . ▁ ▁The ▁one - point ▁compact ification ▁of ▁R ▁is ▁not ▁simply ▁connected ▁( even ▁though ▁R ▁is ▁simply ▁connected ). ▁ ▁The ▁long ▁line ▁L ▁is ▁simply ▁connected , ▁but ▁its ▁compact ification , ▁the ▁extended ▁long ▁line ▁L * ▁is ▁not ▁( since ▁it ▁is ▁not ▁even ▁path ▁connected ). ▁ ▁Properties ▁A ▁surface ▁( two - dimensional ▁topological ▁manifold ) ▁is ▁simply ▁connected ▁if ▁and ▁only ▁if ▁it ▁is ▁connected ▁and ▁its ▁genus ▁( the ▁number ▁of ▁handles ▁of ▁the ▁surface ) ▁is ▁ 0 . ▁ ▁A ▁universal ▁cover ▁of ▁any ▁( su itable ) ▁space ▁X ▁is ▁a ▁simply