Graduate Courses

One of the following topics is taught per year: Dimension Theory: Historical review of Dimension Theory, Small and large inductive dimension, Cover dimension, The basic theorems of n-dimensional spaces (imbedding, union, product and compactification), Universal spaces, Euclidean spaces and Hilbert cube, Posets and Dimension Theory, Dimension theory in Graph Theory, Dynamical systems and Dimension Theory (Carathéodory Dimension, Hausdorff Dimension and Box Dimension). Algebraic Topology: Topological quotient space, Homotopy, Fundamental group, Lifting of mappings, Covering spaces, Covering transformations, Homology groups. Brouwer's fixed point theorem, Borsuk-Ulam theorem, Classification of compact surfaces, Euler-Poincaré characteristic. Special topics in General Topology: Function spaces, Topologies on the sets of open and closed subsets of a topological space, Projective Limits, Čech-Stone compactifications and Wallman extensions, Čech-complete spaces, Paracompact topological spaces, Metricizability theorems of a topological space, Uniform spaces, Proximity spaces. Introduction to Topological Groups and Topological Vector Spaces: Basic definitions and examples in topological groups, Mappings between topological groups, Separability axioms, Subgroups of topological groups, Product of topological groups, Continuity and compactness, Action of topological groups, Basic definitions and examples of the theory of topological vector spaces, Subspaces - products - quotient spaces, Convex and bounded sets, Seminorms and norms, Topological linear spaces with norm, Metrification of topological linear spaces, Locally convex spaces, Continuous linear mappings, Dual spaces. Continuum theory: Examples of continua and nested continua constructions, Products of spaces, Inverse limits of continua and Anderson-Choquet's implantation theorem, Quotient spaces of continua and Upper-semicontinuous partitions. Limits of sets LimInf, LImSup, Lim and convergence theorems, Boundary impact theorems, Convergence subcontinua and its existence theorems, Peano continua (locally connected continuous), S property and Hach-Mazurkiewicz's theorem, The concept of curve and classification of curves, Graphs and Kuratowski's theorem for graphs, Reference to Jordan's theorem and Euler's formula, Dendrites and their basic properties.