1st semester 
20 credits / 30 lectures

The course will introduce basic concepts and ideas from the field of topology. It will try to give some fundamental notions in algebraic topology and geometric topology (which often investigates the connectivity properties of topological spaces by making use of methods from algebra). Some of the topics which will be covered are listed below:

1. Various examples of non-equivalent topologies on the real line;
2. Elementary properties of metric spaces;
3. Basis and local basis for a topological space;
4. Axioms of countability and behaviour of the line of Sorgenfrey;
5. Continuous functions in metric spaces;
6. Examples of induced topologies and quotient topologies;
7. Actions of groups on topological spaces;
8. Lens spaces;
9. Hausdorff property and actions of groups;
10. Product topologies and embeddings in the cube of Tychonoff;
11. Regularity, metrizability and normality in topological spaces;
12. The Lemma of Jones and Lemma of Urysohn;
13. Compact spaces and Lindeloeff spaces;
14. The theorem of Tychonoff and some generalizations;
15. Characterizations of connected spaces;
16. Examples of connected spaces which are not path connected;
17. Connected components and sets which break the connectivity;
18. First notions of algebraic topology;
19. Properly discontinuous actions and fundamental groups;
20. The theorem of Seifert and Van Kampen.