, ▁elect rical ▁engineer ▁and ▁trans it ▁manager . ▁ ▁References ▁ ▁Category : Comm un ities ▁in ▁Per th ▁County , ▁Ontario <0x0A> </s> ▁In ▁topology , ▁a ▁topological ▁space ▁is ▁called ▁simply ▁connected ▁( or ▁ 1 - connected , ▁or ▁ 1 - simp ly ▁connected ) ▁if ▁it ▁is ▁path - connected ▁and ▁every ▁path ▁between ▁two ▁points ▁can ▁be ▁continu ously ▁transformed ▁( int uit ively ▁for ▁embedded ▁spaces , ▁stay ing ▁within ▁the ▁space ) ▁into ▁any ▁other ▁such ▁path ▁while ▁pres erving ▁the ▁two ▁end points ▁in ▁question . ▁The ▁fundamental ▁group ▁of ▁a ▁topological ▁space ▁is ▁an ▁indicator ▁of ▁the ▁failure ▁for ▁the ▁space ▁to ▁be ▁simply ▁connected : ▁a ▁path - connected ▁topological ▁space ▁is ▁simply ▁connected ▁if ▁and ▁only ▁if ▁its ▁fundamental ▁group ▁is ▁trivial . ▁ ▁Definition ▁and ▁equivalent ▁form ulations ▁ ▁A ▁topological ▁space ▁X ▁is ▁called ▁simply ▁connected ▁if ▁it ▁is ▁path - connected ▁and ▁any ▁loop ▁in ▁X ▁defined ▁by ▁f ▁: ▁S 1 ▁→ ▁X ▁can ▁be ▁contract ed ▁to ▁a ▁point : ▁there ▁exists ▁a ▁continuous ▁map ▁F ▁: ▁D 2 ▁→ ▁X ▁such ▁that ▁F ▁restricted ▁to ▁S 1 ▁is ▁f . ▁Here , ▁S 1 ▁and ▁D 2 ▁denotes ▁the ▁unit ▁circle ▁and ▁closed ▁unit ▁disk ▁in ▁the ▁E uclidean ▁plane ▁respectively . ▁ ▁An ▁equivalent ▁form ulation ▁is ▁this : ▁X ▁is ▁simply ▁connected ▁if ▁and ▁only ▁if ▁it ▁is ▁path - connected , ▁and ▁whenever ▁p ▁: ▁[ 0 , 1 ] ▁→