General Topology

The objective of the course is to learn the basic concepts and methods for the study of topological spaces. Also to know how to apply these methods (as well as the previously learned topological and analytical methods) in this general context, especially in the study of quotient spaces.

1. Topological spaces. (4 CLE + 2 CLIL)
Topological spaces. Metric spaces. Interior, closure and boundary. Neighbourhoods. Bases.

2. Continuity. (3 CLE + 2 CLI)
Continuity. Induced topology. Open and closed maps. Homeomorphisms.

3. New constructions. (8 CLE +5 CLIL)
Subspaces. Sum and product spaces. Identifications and quotient spaces. Subspaces and quotient spaces. Group actions and orbit spaces.

4. Separation and countability. (5 CLE + 2 CLIL)
Hausdorff spaces. Hausdorff separation property for quotient spaces. Normal spaces. First-countable spaces. Convergence and closedness. Second-countable spaces. Lindelöf's Theorem.

5. Compactness. (8 CLE + 3 CLIL)
Compact spaces. Tychonoff's Theorem. Hausdorff compact spaces. Compactness in metric spaces. Local compactness.

Basic bibliography
Armstrong M. A., Topología básica. Editorial Reverté. Barcelona, 1987.
Dugundji J., Topology. Allyn and Bacon. Boston, 1966.
Munkres J. R., Topología. Prentice-Hall. Madrid, 2002. Accessible on-line.
Willard S., General Topology. Addison-Wesley. Reading, 1970.

Complementary bibliography
Adams C. and Franzosa R., Introduction to Topology: Pure and Applied. Pearson. 2007
Bourbaki N., Éléments de Mathématique. Topologie générale, chapitres 1 à 4. C.C.L.S, Paris, 1971.
Hu S.T., Elements of General Topology. Holden-Day. San Francisco, 1969.
Krantz S. G., Essentials of Topology with Applications. CRC Press, Boca Raton, 2010.
Masa X.M., Topoloxía Xeral. Manuais Universitarios 1, USC, 1999.
Sutherland W.A., Introduction to metrics and topological spaces. Clarendon Press, Oxford, 1975.

In addition to achieve the general and transverse competences taken up in the memory of the degree, this subject will allow the student to get the following specific competences:
CE1 - To understand and use mathematical language;
CE2 - To know rigorous proofs of some classical theorems in different areas of mathematics;
CE3 - To devise demonstrations of mathematical results, formulate conjectures and imagine strategies to confirm or refute them;
CE4 - To identify errors in faulty reasoning, proposing demonstrations or counterexamples;
CE5 - To assimilate the definition of a new mathematical object, and to be able to use it in different contexts;
CE6 - To identify the abstract properties and material facts of a problem, distinguishing them from those purely occasional or incidental.

2 lectures and 1 problem-based learning session per week. A periodic control of training will be done by means of the resolution of exercices and problems, and also the proposal of collective or individual works.

The qualification of each student will be through continuous evaluation and the completion of a final test on the dates set in the official calendar of the Faculty.

The continuous evaluation will represent 30% of the final grade. It will be carried out throughout the course based on the participation of each student in class, resolution and/or presentation of problems proposed in the different bulletins, and two written controls, which will have a weight of 60% in the EC.

-The final exam will consist of a written test with a theoretical part, which may include the definition of concepts, the statement of results and the total or partial proof of them, and a practical part consisting of the resolution of problems and exercises similar to those solved in the laboratory classes. It will represent 70% of the final grade.

-The grade obtained in the continuous evaluation will be applicable in each of the two opportunities of the same academic year (second semester and July). If the student does not take the exam set by the faculty in either of the two opportunities, he/she will have the grade of "Not presented", even if he/she has participated in the continuous evaluation.

-The exams of each group will be independent but similar.

-According to the approved regulations, the final grade will be at least the grade of the final exam.

Hours of classroom work:
Lectures 28
Interactive laboratory classes 14
Tutorials in very small groups or individualized tutorials 2
Total hours of classroom work 44

Student work hours
Theoretical and practical study related to face-to-face teaching 49
Preparation of exercises and the written test 19
Total hours of personal work 60