Editing Convex set (section)

=== Convexity spaces ===
The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as [[axiom]]s.

Given a set {{mvar|X}}, a '''convexity''' over {{mvar|X}} is a collection {{math|''𝒞''}} of subsets of {{mvar|X}} satisfying the following axioms:<ref name="Soltan"/><ref name="Singer"/><ref name="vanDeVel" >{{cite book|last=van De Vel|first=Marcel L. J.|title=Theory of convex structures|series=North-Holland Mathematical Library|publisher=North-Holland Publishing Co.|location=Amsterdam|year= 1993|pages=xvi+540|isbn=0-444-81505-8|mr=1234493}}</ref>

#The empty set and {{mvar|X}} are in {{math|''𝒞''}}.
#The intersection of any collection from {{math|''𝒞''}} is in {{math|''𝒞''}}.
#The union of a [[Total order|chain]] (with respect to the [[inclusion relation]]) of elements of {{math|''𝒞''}} is in {{math|''𝒞''}}.

The elements of {{math|''𝒞''}} are called convex sets and the pair {{math|(''X'', ''𝒞'')}} is called a '''convexity space'''. For the ordinary convexity, the first two axioms hold, and the third one is trivial.

For an alternative definition of abstract convexity, more suited to [[discrete geometry]], see the ''convex geometries'' associated with [[antimatroid]]s.