▁exist ▁a ▁region ▁included ▁in ▁all ▁of ▁the ▁regions ▁included ▁in ▁G . ▁ ▁Int uit ively , ▁an ▁abstract ive ▁class ▁defines ▁a ▁geomet rical ▁entity ▁whose ▁dimensional ity ▁is ▁less ▁than ▁that ▁of ▁the ▁inclusion ▁space . ▁For ▁example , ▁if ▁the ▁inclusion ▁space ▁is ▁the ▁E uclidean ▁plane , ▁then ▁the ▁corresponding ▁abstract ive ▁classes ▁are ▁points ▁and ▁lines . ▁ ▁In clusion - based ▁point - free ▁geometry ▁( h ence for th ▁" point - free ▁geometry ") ▁is ▁essentially ▁an ▁ax iom at ization ▁of ▁Sim ons ' s ▁( 1 9 8 7 : ▁ 8 3 ) ▁system ▁W . ▁In ▁turn , ▁W ▁formal izes ▁a ▁theory ▁in ▁White head ▁( 1 9 1 9 ) ▁whose ▁ax i oms ▁are ▁not ▁made ▁explicit . ▁Point - free ▁geometry ▁is ▁W ▁with ▁this ▁defect ▁rep a ired . ▁Sim ons ▁( 1 9 8 7 ) ▁did ▁not ▁repair ▁this ▁defect , ▁instead ▁propos ing ▁in ▁a ▁foot note ▁that ▁the ▁reader ▁do ▁so ▁as ▁an ▁exercise . ▁The ▁primitive ▁relation ▁of ▁W ▁is ▁Pro per ▁Part , ▁a ▁strict ▁partial ▁order . ▁The ▁theory ▁of ▁White head ▁( 1 9 1 9 ) ▁has ▁a ▁single ▁primitive ▁binary ▁relation ▁K ▁defined ▁as ▁x K y ▁ ↔ ▁y < x . ▁Hence ▁K ▁is ▁the ▁con verse ▁of ▁Pro per ▁Part . ▁Sim ons ' s ▁W P 1 ▁as ser ts ▁that ▁Pro per ▁Part ▁is ▁ir ref lex ive ▁and ▁so