▁condition ▁is ▁necessary ▁for ▁most ▁of ▁the ▁theory ▁of ▁covering ▁spaces , ▁including ▁the ▁existence ▁of ▁a ▁universal ▁cover ▁and ▁the ▁Gal ois ▁correspond ence ▁between ▁covering ▁spaces ▁and ▁sub groups ▁of ▁the ▁fundamental ▁group . ▁ ▁Most ▁“ nice ” ▁spaces ▁such ▁as ▁manif olds ▁and ▁C W ▁complex es ▁are ▁semi - loc ally ▁simply ▁connected , ▁and ▁topological ▁spaces ▁that ▁do ▁not ▁satisfy ▁this ▁condition ▁are ▁considered ▁somewhat ▁path ological . ▁The ▁standard ▁example ▁of ▁a ▁non - sem i - loc ally ▁simply ▁connected ▁space ▁is ▁the ▁Hawai ian ▁ear ring . ▁ ▁Definition ▁A ▁space ▁X ▁is ▁called ▁semi - loc ally ▁simply ▁connected ▁if ▁every ▁point ▁in ▁X ▁has ▁a ▁neighborhood ▁U ▁with ▁the ▁property ▁that ▁every ▁loop ▁in ▁U ▁can ▁be ▁contract ed ▁to ▁a ▁single ▁point ▁within ▁X ▁( i . e .   every ▁loop ▁in ▁U ▁is ▁null hom ot op ic ▁in ▁X ). ▁The ▁neighborhood ▁U ▁need ▁not ▁be ▁simply ▁connected : ▁though ▁every ▁loop ▁in ▁U ▁must ▁be ▁contract ible ▁within ▁X , ▁the ▁contra ction ▁is ▁not ▁required ▁to ▁take ▁place ▁inside ▁of ▁U . ▁For ▁this ▁reason , ▁a ▁space ▁can ▁be ▁semi - loc ally ▁simply ▁connected ▁without ▁being ▁locally ▁simply ▁connected ▁( see ▁# Ex amples ). ▁ ▁Equ ivalent ▁to ▁this ▁definition , ▁a ▁space ▁X ▁is ▁semi - loc ally ▁simply ▁connected ▁if ▁every ▁point ▁in ▁X ▁has ▁a ▁neighborhood ▁U ▁for ▁which ▁the ▁hom omorphism ▁from ▁the ▁fundamental ▁group ▁of ▁U ▁to ▁the