A characterization of complete Boolean algebras (Q2725210)

A convexity space is a set \(X\) endowed with a collection \(\mathcal C\subset\mathcal P(X)\) containing \(\emptyset\), \(X\), closed under arbitrary intersections and closed under the unions of chains. Elements of the collection \(\mathcal C\) are called convex subsets of \(X\). A convexity space \(X\) is \(S_4\) provided any two disjoint convex sets \(A,B\subset X\) can be separated by a halfspace (i.e. a convex set with convex complement) \(H\subset X\) in the sense that \(A\subset H\) and \(B\subset X\setminus H\). A convexity space \(X\) is \(S_3\) if any one-point subset of \(X\) is convex and any convex subset \(A\subset X\) can be separated from any point \(x\in X\setminus A\) by a halfspace. NEWLINENEWLINENEWLINEThe main result of the paper states that an \(S_3\) convexity space \(X\) is isomorphic to a complete Boolean algebra (endowed with the natural convexity structure) if and only if \(X\) has the following extension property: every convexity preserving map \(f:B\to X\) defined on a convex subset \(B\) of an \(S_4\) convexity space \(A\) can be extended to a convexity preserving map \(\overline f:A\to X\). (A map \(f:X\to Y\) between convexity spaces is called convexity preserving if the preimage \(f^{-1}(C)\) of any convex subset \(C\subset Y\) is convex in \(X\).) NEWLINENEWLINENEWLINEAs a corollary the author obtains a classical extension theorem of Sikorski asserting that any homomorphism \(h:K\to B\) from a sublattice \(K\) of a distributive lattice \(L\) into a complete Boolean algebra \(B\) can be extended to a homomorphism \(\overline h:L\to B\).