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.

• the DVI and PDF file of my thesis (59 pages, 200kb/400kb)
• and
• the DVI and PDF file of my Master's Thesis (warning: it is in Polish and contains some non-standard fonts, 29 pages, 100kb/300kb).

##### Abstract Convex Structures in Topology and Set Theory, dissertation
###### ABSTRACT

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 [1] and in a more recent monograph of M. van de Vel [2]. In a general setting, the axioms of convexity are the following:

1. The empty set and the whole space are convex;
2. The intersection of a nonempty collection of convex sets is convex;
3. The union of a chain of convex sets is convex.
Clearly, usual convex sets have properties (1) - (3) but there are many other collections of sets, coming from various types of mathematical objects, that satisfy conditions (1) - (3). Probably one of the most interesting are convexities in metric spaces and graphs and convexities in lattices. Convex structures appear naturally in topology, especially in the theory of supercompact spaces. There are also considered morphisms between convexities, called convexity preserving mappings. This notion allows us to compare different convexities and to define isomorphisms, embeddings, products, etc.

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

###### References

1. V.P. Soltan, Introduction to the Axiomatic Theory of Convexity, (Russian) Shtiinca, Kishinev 1984.
2. M. van de Vel, Theory of Convex Structures, North-Holland, Amsterdam 1993.