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.

Topics:

• 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.