Axiomatic theory of convexity

The axiomatic construction of the theory of convexity proceeds from an arbitrary set $M$ and a mapping $l:M^2\to2^M$, i.e., from a pair $(M,l)$. It is shown that such a space of a certain type is domain finite. A condition is given which, for such spaces, implies join-hull commutativity. A connection is established between the Carathéodory number and join-hull commutativity. Conditions are given which imply a separation property of the space $(M,l)$. Convexity spaces which are domain finite are characterized.