▁fundamental ▁group ▁of ▁X , ▁induced ▁by ▁the ▁inclusion ▁map ▁of ▁U ▁into ▁X , ▁is ▁trivial . ▁ ▁Most ▁of ▁the ▁main ▁the or ems ▁about ▁covering ▁spaces , ▁including ▁the ▁existence ▁of ▁a ▁universal ▁cover ▁and ▁the ▁Gal ois ▁correspond ence , ▁require ▁a ▁space ▁to ▁be ▁path - connected , ▁locally ▁path - connected , ▁and ▁semi - loc ally ▁simply ▁connected , ▁a ▁condition ▁known ▁as ▁un loop able ▁( d é la ç able ▁in ▁French ). ▁In ▁particular , ▁this ▁condition ▁is ▁necessary ▁for ▁a ▁space ▁to ▁have ▁a ▁simply ▁connected ▁covering ▁space . ▁ ▁Ex amples ▁ ▁A ▁simple ▁example ▁of ▁a ▁space ▁that ▁is ▁not ▁semi - loc ally ▁simply ▁connected ▁is ▁the ▁Hawai ian ▁ear ring : ▁the ▁union ▁of ▁the ▁circles ▁in ▁the ▁E uclidean ▁plane ▁with ▁cent ers ▁( 1 / n , ▁ 0 ) ▁and ▁rad ii ▁ 1 / n , ▁for ▁n ▁a ▁natural ▁number . ▁Give ▁this ▁space ▁the ▁sub space ▁topology . ▁Then ▁all ▁neighborhood s ▁of ▁the ▁origin ▁contain ▁circles ▁that ▁are ▁not ▁null hom ot op ic . ▁ ▁The ▁Hawai ian ▁ear ring ▁can ▁also ▁be ▁used ▁to ▁construct ▁a ▁semi - loc ally ▁simply ▁connected ▁space ▁that ▁is ▁not ▁locally ▁simply ▁connected . ▁In ▁particular , ▁the ▁cone ▁on ▁the ▁Hawai ian ▁ear ring ▁is ▁contract ible ▁and ▁therefore ▁semi - loc ally ▁simply ▁connected , ▁but ▁it ▁is ▁clearly ▁not ▁locally ▁simply ▁connected . ▁ ▁Top ology ▁of ▁fundamental ▁group ▁In