*Instructor:* Dr. Gábor MOUSSONG

*Text:* M. A. Armstrong, *Basic Topology*, Springer Undergraduate
Texts in Mathematics, 1983. (available digitally here)

*Prerequisite:*
Firm knowledge of some standard concepts of first-year calculus (like limits and
continuity, manipulating with sets and fuctions) is indispensable. Rudiments of
group theory (not much more than uderstanding what the words group, subgroup,
homomorphism, isomorphism mean) will also be necessary in the second half of the
course.

*Course description:*
Beginner's introduction to the fundamentals of topology. The lecture roughly
follows the first half of Armstrong's book. The first part of the course deals
with abstract point-set topology, and he second part introduces some of the more
geometric and algebraic ideas.

*Topics:*

- Introduction: An informal presentation of some of the motivating questions and ideas coming from geometry and calculus. Metric spaces.
- Basic definitions: Topological spaces, properties of open and closed sets. Continuous maps, homeomorphisms, topological invariants. Limits, Hausdorff spaces.
- Connectedness and compactness of topological spaces.
- Cut-and-paste topology: Quotients, identification spaces, 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, applications.