Some properties of ultrametric spaces and their relations to category theory and computer science

The Thesis is devoted to the theory of ultrametric spaces, i.e., metric spaces where the strengthened triangle inequality holds: d( x, z) ≤ max [d(x, y), d(y, z)]. Particular examples of these spaces have been known for more that 100 years since Kurt Hensel defined rings Z_p and fields Q _p of p-adic numbers in number theory (1894, 1904), René Baire introduced the Baire space in real analysis (1906), and Felix Hausdorff studied “nichtarchimedische Metrik” in topology (1914). During the last two decades the theory of ultrametric spaces, has found a number of strong relations to Euclidean geometry and geometry of Hilbert space, Lebesgue measure and integral theory, lattice theory and theory of Boolean algebras, set theory and foundations, category theory and computer science. The theory of ultrametric spaces serves as a base for p-adic analysis, p-adic functional analysis, and the theory of p-adic analytic manifolds.; Most of the results of the Thesis are published in my papers listed in the Bibliography, in particular, in “Finite ultrametric spaces and computer science” - “Categorical Perspectives” ed. Jürgen Koslowski, Austin Melton (Trends in Mathematics, v. 16), Birkhäuser Verlag, Boston-Basel-Berlin, 2001, p.219–241.; I presented some results at the following international mathematical conferences and symposia: (1) Prague, Czech Republic (at 8th Prague Topological Symposium, August 1996), (2) North Bay, Canada (at 14th Summer Conference on General Topology and its Applications, August 1997), (3) Milazzo, Italy (at 14th International Congress on Topology, September 1997), (4) Kent, USA (at International Conference on Georgian-Influenced Mathematics, August 1998).; is composed of four Chapters each of which is devoted to a particular aspect of the theory. Chapter 1. Metrically universal ultrametric spaces. Chapter 2. Categorical operations and imbedding theorems. Chapter 3. General metric spaces as images and pre-images of ultrametric spaces. Chapter 4. Finite ultrametric spaces, rationalization of ultrametrics, and applications to Computer Science.