I finished my Ph.D. thesis, entitled Abstract Convex Structures in Topology and Set Theory, on Summer 1999; the defense was on May 30, 2000. Below you find the abstract of my dissertation.
Here you can download:
Abstract convexity theory is a branch of mathematics dealing with set-theoretic structures satisfying axioms similar to that usual convex sets fulfill. Here, by "usual convex sets" we mean convex sets in real linear spaces. First papers on axiomatic convexities come from early fifties, a complete bibliography can be found in a book of V.P. Soltan  and in a more recent monograph of M. van de Vel . In a general setting, the axioms of convexity are the following:
The dissertation deals with various properties related to separation and extending of convexity preserving maps. By "separation" we mean the property that certain two maps from fixed two classes can be separated (by means of a suitable partial order) by a third map which belongs to both classes simultaneously. It appears that complete Boolean algebras (as convexity spaces) play a fundamental role in developing these properties. We also investigate topological convexity spaces, specifically compact median spaces. We are interested in a problem of continuous convexity preserving extending maps. Our results are applied for finding some linearly ordered subspaces of compact median spaces.
Katowice, August 1999