*Instructor:* Ágnes SZILARD PhD

*Text:* Your class notes and online notes (will be available on the class homepage).

*Prerequisites:*
Solid knowledge of calculus (limits and
continuity, manipulating with sets and fuctions) is indispensable. 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.

*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: retracts and deformation retracts, covering spaces. If time permits: properly discontinuous group actions.