Course description: Beginner's introduction to fundamental concepts of topology. The first part of the course deals with abstract point-set topology, and the second part introduces some more geometric and algebraic ideas.

Topics:

• Introduction: Some motivating questions coming from geometry and calculus. The concept of continuity in metric spaces.
• Basic definitions: Topological spaces, open and closed sets. Continuous maps, homeomorphisms, topological invariants. Limits, Hausdorff spaces.
• Constructions: Subspaces, products, quotients.
• Connectedness and compactness of topological spaces.
• Cut-and-paste topology: Gluings, constructions of surfaces. Sketch proof of the classification theorem of closed surfaces.
• Homotopy: Homotopic maps, homotopy type of spaces, homotopy invariants.
• The fundamental group: Definitions and methods of calculation, some applications.