▁value ▁theorem , ▁there ▁is ▁some ▁point ▁x 0 ▁with ▁g ( x 0 ) ▁= ▁ 0 , ▁which ▁is ▁to ▁say ▁that ▁f ( x 0 ) ▁− ▁x 0 ▁= ▁ 0 , ▁and ▁so ▁x 0 ▁is ▁a ▁fixed ▁point . ▁ ▁The ▁open ▁interval ▁does ▁not ▁have ▁the ▁fixed - point ▁property . ▁The ▁mapping ▁f ( x ) ▁= ▁x 2 ▁has ▁no ▁fixed ▁point ▁on ▁the ▁interval ▁( 0 , 1 ). ▁ ▁The ▁closed ▁disc ▁The ▁closed ▁interval ▁is ▁a ▁special ▁case ▁of ▁the ▁closed ▁disc , ▁which ▁in ▁any ▁finite ▁dimension ▁has ▁the ▁fixed - point ▁property ▁by ▁the ▁Br ou wer ▁fixed - point ▁theorem . ▁ ▁Top ology ▁A ▁ret ract ▁A ▁of ▁a ▁space ▁X ▁with ▁the ▁fixed - point ▁property ▁also ▁has ▁the ▁fixed - point ▁property . ▁ ▁This ▁is ▁because ▁if ▁ ▁is ▁a ▁retra ction ▁and ▁ ▁is ▁any ▁continuous ▁function , ▁then ▁the ▁composition ▁ ▁( where ▁ ▁is ▁inclusion ) ▁has ▁a ▁fixed ▁point . ▁ ▁That ▁is , ▁there ▁is ▁ ▁such ▁that ▁. ▁ ▁Since ▁ ▁we ▁have ▁that ▁ ▁and ▁therefore ▁▁ ▁A ▁topological ▁space ▁has ▁the ▁fixed - point ▁property ▁if ▁and ▁only ▁if ▁its ▁identity ▁map ▁is ▁universal . ▁ ▁A ▁ ▁product ▁of ▁spaces ▁with ▁the ▁fixed - point ▁property ▁in ▁general ▁fails ▁to ▁have ▁the ▁fixed - point ▁property ▁even ▁if ▁one ▁of ▁the ▁spaces ▁is ▁the ▁closed ▁real ▁interval . ▁ ▁The ▁ ▁F PP ▁is ▁a