Introduction to Topology

  • Instructor: Ágnes SZILARD PhD
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  • Prerequisites: Solid knowledge of calculus (limits and continuity, manipulating with sets and fuctions) is indispensable. An introductory course in real analysis is suggested (not required). Rudiments of group theory (uderstanding of what a group, homomorphism, isomorphism is, as well as familiarity with basic groups such as cyclic groups, Z, ZxZ) will also be necessary in the second half of the course.
  • Text: notes will be posted

Course description: This is a standard introductory course the goal of which is to get acquainted with the basic notions of the field. Thus we start with point-set topology and the study of topological spaces, in particular metric spaces, continuity, connectedness, compactness. The machinery developed will allow us to consider one of the major theorems of topology: the classification of compact, connected surfaces. In the second half of the course we get a taste of algebraic topology - the notion of the fundamental group of a topological space will be introduced as well as several ways of computing it. Throughout the course we will study numerous examples and applications.


  • Topological spaces, homeomorphism. First examples. The classification problem and the role of topological invariants.
  • Constructing new topologies from given ones: the subspace, quotient and product topologies.
  • Some topological invariants: the Hausdorff property, compactness, connectedness, path-connectedness.
  • Compact, connected surfaces. Euler characteristic and orientability. The classification theorem of compact connected surfaces.
  • The fundamental group. Intuitive examples.
  • Methods to calculate the fundamental group: covering spaces, retracts and deformation retracts.
  • If time permits: properly discontinuous group actions.