▁X ▁and ▁q ▁: ▁[ 0 , 1 ] ▁→ ▁X ▁are ▁two ▁paths ▁( i . e .: ▁continuous ▁maps ) ▁with ▁the ▁same ▁start ▁and ▁endpoint ▁( p ( 0 ) ▁= ▁q ( 0 ) ▁and ▁p ( 1 ) ▁= ▁q ( 1 )), ▁then ▁p ▁can ▁be ▁continu ously ▁de formed ▁into ▁q ▁while ▁keeping ▁both ▁end points ▁fixed . ▁Exp licit ly , ▁there ▁exists ▁a ▁hom ot opy ▁ ▁such ▁that ▁ ▁and ▁. ▁ ▁A ▁topological ▁space ▁X ▁is ▁simply ▁connected ▁if ▁and ▁only ▁if ▁X ▁is ▁path - connected ▁and ▁the ▁fundamental ▁group ▁of ▁X ▁at ▁each ▁point ▁is ▁trivial , ▁i . e . ▁consists ▁only ▁of ▁the ▁identity ▁element . ▁Similarly , ▁ ▁is ▁simply ▁connected ▁if ▁and ▁only ▁if ▁for ▁all ▁points ▁, ▁the ▁set ▁of ▁morph isms ▁ ▁in ▁the ▁fundamental ▁gr ou po id ▁of ▁ ▁has ▁only ▁one ▁element . ▁ ▁In ▁complex ▁analysis : ▁an ▁open ▁subset ▁ ▁is ▁simply ▁connected ▁if ▁and ▁only ▁if ▁both ▁X ▁and ▁its ▁complement ▁in ▁the ▁Riemann ▁sphere ▁are ▁connected . ▁The ▁set ▁of ▁complex ▁numbers ▁with ▁imag inary ▁part ▁strictly ▁greater ▁than ▁zero ▁and ▁less ▁than ▁one , ▁furn ishes ▁a ▁nice ▁example ▁of ▁an ▁un bounded , ▁connected , ▁open ▁subset ▁of ▁the ▁plane ▁whose ▁complement ▁is ▁not ▁connected . ▁It ▁is ▁never theless ▁simply ▁connected . ▁It ▁might ▁also ▁be ▁worth ▁pointing ▁out ▁that ▁a ▁relax ation ▁of ▁the ▁requirement ▁that ▁X ▁be ▁connected ▁leads ▁to ▁an ▁interesting