, ▁are ▁called ▁mere ot op ologies . ▁ ▁C ▁has ▁one ▁primitive ▁relation , ▁binary ▁" connection ," ▁denoted ▁by ▁the ▁prefix ed ▁predicate ▁letter ▁C . ▁That ▁x ▁is ▁included ▁in ▁y ▁can ▁now ▁be ▁defined ▁as ▁x ≤ y ▁ ↔ ▁ ∀ z [ C z x → C zy ]. ▁Un like ▁the ▁case ▁with ▁inclusion ▁spaces , ▁connection ▁theory ▁enables ▁defining ▁" non - t ang ential " ▁inclusion , ▁a ▁total ▁order ▁that ▁enables ▁the ▁construction ▁of ▁abstract ive ▁classes . ▁Ger la ▁and ▁Mir anda ▁( 2 0 0 8 ) ▁argue ▁that ▁only ▁thus ▁can ▁mere ot op ology ▁un ambigu ously ▁define ▁a ▁point . ▁ ▁The ▁ax i oms ▁C 1 - C 6 ▁below ▁are , ▁but ▁for ▁number ing , ▁those ▁of ▁Def . ▁ 3 . 1 ▁in ▁Ger la ▁and ▁Mir anda ▁( 2 0 0 8 ). ▁ ▁C ▁is ▁ref lex ive . ▁C . 1 . ▁C 1 . ▁▁ ▁C ▁is ▁symmetric . ▁C . 2 . ▁C 2 . ▁▁▁ ▁C ▁is ▁ext ensional . ▁C . 1 1 . ▁C 3 . ▁▁▁▁▁ ▁All ▁regions ▁have ▁proper ▁parts , ▁so ▁that ▁C ▁is ▁an ▁atom less ▁theory . ▁P . 9 . ▁C 4 . ▁▁▁▁▁ ▁Given ▁any ▁two ▁regions , ▁there ▁is ▁a ▁region ▁connected ▁to ▁both ▁of ▁them . ▁C 5 . ▁▁▁▁ ▁All ▁regions ▁have ▁at ▁least ▁two ▁un connected ▁parts . ▁C . 1 4 . ▁C 6