▁point ▁property ▁if ▁every ▁increasing ▁function ▁on ▁P ▁has ▁a ▁fixed ▁point . ▁ ▁Definition ▁Let ▁A ▁be ▁an ▁object ▁in ▁the ▁concrete ▁category ▁C . ▁ ▁Then ▁A ▁has ▁the ▁fixed - point ▁property ▁if ▁every ▁morph ism ▁( i . e ., ▁every ▁function ) ▁ ▁has ▁a ▁fixed ▁point . ▁ ▁The ▁most ▁common ▁usage ▁is ▁when ▁C   =   Top ▁is ▁the ▁category ▁of ▁topological ▁spaces . ▁ ▁Then ▁a ▁topological ▁space ▁X ▁has ▁the ▁fixed - point ▁property ▁if ▁every ▁continuous ▁map ▁ ▁has ▁a ▁fixed ▁point . ▁ ▁Ex amples ▁ ▁Sing let ons ▁In ▁the ▁category ▁of ▁sets , ▁the ▁objects ▁with ▁the ▁fixed - point ▁property ▁are ▁precisely ▁the ▁sing let ons . ▁ ▁The ▁closed ▁interval ▁The ▁closed ▁interval ▁[ 0 , 1 ] ▁has ▁the ▁fixed ▁point ▁property : ▁Let ▁f : ▁[ 0 , 1 ] ▁→ ▁[ 0 , 1 ] ▁be ▁a ▁continuous ▁mapping . ▁If ▁f ( 0 ) ▁= ▁ 0 ▁or ▁f ( 1 ) ▁= ▁ 1 , ▁then ▁our ▁mapping ▁has ▁a ▁fixed ▁point ▁at ▁ 0 ▁or ▁ 1 . ▁If ▁not , ▁then ▁f ( 0 ) ▁> ▁ 0 ▁and ▁f ( 1 ) ▁− ▁ 1 ▁< ▁ 0 . ▁Thus ▁the ▁function ▁g ( x ) ▁= ▁f ( x ) ▁− ▁x ▁is ▁a ▁continuous ▁real ▁val ued ▁function ▁which ▁is ▁positive ▁at ▁x ▁= ▁ 0 ▁and ▁negative ▁at ▁x ▁= ▁ 1 . ▁By ▁the ▁intermediate