Separation properties of convexity spaces. (Q1865875)

Remember that by a convexity structure in a set \(X\) we mean a family \(\mathcal C\) of subsets of \(X\) which is closed under intersections and under the unions of chains, and which contains \(\emptyset\) and \(X\). The author considers two convexity structures \(\mathcal G\) and \(\mathcal H\) in a set \(X\). The main theorem of this paper presents a condition equivalent to the following separation condition: if \(A \in \mathcal G\) and \(B\in \mathcal H\) are disjoint, then there are disjoint sets \(G \in \mathcal G\) containing \(A\), and \(H \in \mathcal H\) containing \(B\), such that \(G\cup H =X\).