▁expl oration ▁of ▁open ▁subsets ▁of ▁the ▁plane ▁with ▁connected ▁extended ▁complement . ▁For ▁example , ▁a ▁( not ▁necessarily ▁connected ) ▁open ▁set ▁has ▁connected ▁extended ▁complement ▁exactly ▁when ▁each ▁of ▁its ▁connected ▁components ▁are ▁simply ▁connected . ▁ ▁Inform al ▁discussion ▁ ▁Inform ally , ▁an ▁object ▁in ▁our ▁space ▁is ▁simply ▁connected ▁if ▁it ▁consists ▁of ▁one ▁piece ▁and ▁does ▁not ▁have ▁any ▁" hol es " ▁that ▁pass ▁all ▁the ▁way ▁through ▁it . ▁For ▁example , ▁neither ▁a ▁d ough nut ▁nor ▁a ▁coffee ▁cup ▁( with ▁handle ) ▁is ▁simply ▁connected , ▁but ▁a ▁h ollow ▁rub ber ▁ball ▁is ▁simply ▁connected . ▁In ▁two ▁dimensions , ▁a ▁circle ▁is ▁not ▁simply ▁connected , ▁but ▁a ▁disk ▁and ▁a ▁line ▁are . ▁Sp aces ▁that ▁are ▁connected ▁but ▁not ▁simply ▁connected ▁are ▁called ▁non - simp ly ▁connected ▁or ▁multiply ▁connected . ▁ ▁The ▁definition ▁only ▁rules ▁out ▁handle - sh aped ▁holes . ▁A ▁sphere ▁( or , ▁equival ently , ▁a ▁rub ber ▁ball ▁with ▁a ▁h ollow ▁center ) ▁is ▁simply ▁connected , ▁because ▁any ▁loop ▁on ▁the ▁surface ▁of ▁a ▁sphere ▁can ▁contract ▁to ▁a ▁point , ▁even ▁though ▁it ▁has ▁a ▁" hole " ▁in ▁the ▁h ollow ▁center . ▁The ▁stronger ▁condition , ▁that ▁the ▁object ▁has ▁no ▁holes ▁of ▁any ▁dimension , ▁is ▁called ▁contract ibility . ▁ ▁Ex amples ▁▁ ▁The ▁E uclidean ▁plane ▁R 2 ▁is ▁simply ▁connected , ▁but ▁R 2 ▁minus ▁the ▁origin ▁( 0 ,