The topics are divided into two modules:

- General Topology
- Topological spaces, open and closed sets, continuous maps, homeomorphism.
- Connected, path connected, and locally connected spaces. Connected subspaces of \(\mathbb{R}\).
- Compact and locally compact spaces. Compact subspaces of \(\mathbb{R}^n\). Compactness in metric spaces.
- Countability axioms.
- Separation axioms. The Urysohn Lemma.
- The Urysohn Metrization Theorem.
- The Tychonoff Theorem.
- The Stone-Cech compactification.
- Complete metric spaces. Contraction theorem.
- Functional spaces and their topologies. The Arzela-Ascolli Theorem.
- The Baire property and the Baire Theorem.

- Algebraic Topology
- Homotopy and homotopy type. Deformation retracts and deformation retractions. Contractible spaces.
- The fundamental group. The fundamental groups of the circle. A topological proof of the Fundamental Theorem of Algebra.
- The Seifert-van Kampen Theorem.
- Surfaces and their fundamental groups.
- Covering spaces.
- Exact sequences, 5-lemma. Homology groups, homology exact sequence. Homotopy invariance of homology groups. Excision.
- Homology groups of \(n\)-dimensional sphere. The Brouwer Fixed Point Theorem. The degree of a map \(S^n \to S^n\). Vector fields on \(S^n\).
- The Euler characteristic. Lefschetz Fixed Point Theorem
- Cohomology. The cup product and the cap product. The Borsuk-Ulam Theorem.
- The Universal Coefficient theorem.
- Universal Coefficient Formula, The Künneth Formula.
- Manifolds. Orientation. Homology and cohomology of manifolds. Poincaré Duality. The degree of a map.

**Literature:**

- J. Munkres,
*Topology, 2nd edition*, Prentice Hall - A. Hatcher,
*Algebraic Topology*, Cambridge University Press;

**Additional Resources:**

(posted March 19, 2014)