▁connected ▁space ▁which ▁maps ▁to ▁X ▁via ▁a ▁covering ▁map . ▁ ▁If ▁X ▁and ▁Y ▁are ▁hom ot opy ▁equivalent ▁and ▁X ▁is ▁simply ▁connected , ▁then ▁so ▁is ▁Y . ▁ ▁The ▁image ▁of ▁a ▁simply ▁connected ▁set ▁under ▁a ▁continuous ▁function ▁need ▁not ▁be ▁simply ▁connected . ▁Take ▁for ▁example ▁the ▁complex ▁plane ▁under ▁the ▁exponential ▁map : ▁the ▁image ▁is ▁C ▁- ▁{ 0 }, ▁which ▁is ▁not ▁simply ▁connected . ▁ ▁The ▁notion ▁of ▁simple ▁connected ness ▁is ▁important ▁in ▁complex ▁analysis ▁because ▁of ▁the ▁following ▁facts : ▁▁ ▁The ▁C auch y ' s ▁integral ▁theorem ▁states ▁that ▁if ▁U ▁is ▁a ▁simply ▁connected ▁open ▁subset ▁of ▁the ▁complex ▁plane ▁C , ▁and ▁f ▁: ▁U ▁→ ▁C ▁is ▁a ▁hol omorphic ▁function , ▁then ▁f ▁has ▁an ▁ant ider iv ative ▁F ▁on ▁U , ▁and ▁the ▁value ▁of ▁every ▁line ▁integral ▁in ▁U ▁with ▁integr and ▁f ▁depends ▁only ▁on ▁the ▁end ▁points ▁u ▁and ▁v ▁of ▁the ▁path , ▁and ▁can ▁be ▁computed ▁as ▁F ( v ) ▁- ▁F ( u ). ▁The ▁integral ▁thus ▁does ▁not ▁depend ▁on ▁the ▁particular ▁path ▁connecting ▁u ▁and ▁v . ▁▁ ▁The ▁Riemann ▁mapping ▁theorem ▁states ▁that ▁any ▁non - empty ▁open ▁simply ▁connected ▁subset ▁of ▁C ▁( except ▁for ▁C ▁itself ) ▁is ▁conform ally ▁equivalent ▁to ▁the ▁unit ▁disk . ▁ ▁The ▁notion ▁of ▁simple ▁connected ness ▁is ▁also ▁a ▁cru cial ▁condition ▁in ▁the ▁Po inc ar é ▁conject ure . ▁ ▁See