Instructor: Árpád Tóth

Text: Notes by László Fehér available at http://www.renyi.hu/~lfeher, http:/www.cs.elte.hu/~lfeher, http:/ww.cs.elte.hu/~toth or may be bought at the office
         You may find the book of Munkres (Topology) useful

Prerequisite: Calculus, especially metric spaces, notion of continuity, basics of set theory, definition and basic properties of groups.

Course description:  We start with point set topology, which is an essential part of modern analysis, to be ready for algebraic topology, which is one of the most dramatic topic of modern geometry. Level depends on the background of the class.

Topics:

• Experiments in Topology: Untie knots, Set yourself free, Spinning plates and why did Dirac got the Nobel price, The chess problem the computer couldn't solve?
• The notion of continuity
• Metric spaces and continuous functions
• Open and closed sets
• Equivalent metrics: the notion of topological space
• Examples and constructions
• Translating the experiments into mathematics
• Separation axioms: the zoo
• Compactness
• Surfaces and other manifolds
• How to distinguish them? Euler characteristic, brushing the hedgehog, the coloring problem.
• The curve which fills the square. Why dimension is a topological notion?
• Connectivity
• Homotopy and the fundamental group
• Fundamental group of the circle and the sphere
• The Seifert Van Kampen Theorem
• Applications, back to the experiments
• Prospects: homology, higher homotopies