ECTS credits ECTS credits: 4.5
ECTS Hours Rules/Memories Student's work ECTS: 74.2 Hours of tutorials: 2.25 Expository Class: 18 Interactive Classroom: 18 Total: 112.45
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Geometry and Topology
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable
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.
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.
Munkres J. R., Topología. Prentice-Hall. Madrid, 2002.
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.
Continual assessment based upon the participation of each student at class (30% of the final grade) and final assessment by means of a written test fixed in the calendar of the Faculty (70% of the final grade). The written test will consist on some theoretical questions, that may include definition of concepts, statement of results and total or partial proof of them, and some problems analogous to those solved in the course.
44 hours of lectures
68 further hours' study
Enrique Macías Virgós
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813153
- quique.macias [at] usc.es
- Category
- Professor: University Professor
Fernando Alcalde Cuesta
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813142
- fernando.alcalde [at] usc.es
- Category
- Professor: University Lecturer
Álvaro Carballido Costas
- Department
- Mathematics
- Area
- Geometry and Topology
- alvaro.carballido.costas [at] usc.es
- Category
- Xunta Pre-doctoral Contract
Juan Manuel Lorenzo Naveiro
- Department
- Mathematics
- Area
- Geometry and Topology
- jm.lorenzo [at] usc.es
- Category
- Ministry Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 06 |
| Tuesday | |||
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom 03 |
| 11:00-12:00 | Grupo /CLE_02 | Spanish | Classroom 06 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLE_02 | Spanish | Classroom 06 |
| Thursday | |||
| 09:00-10:00 | Grupo /CLIL_04 | Spanish | Classroom 07 |
| 10:00-11:00 | Grupo /CLIL_03 | Spanish | Classroom 07 |
| 11:00-12:00 | Grupo /CLIL_01 | Spanish | Classroom 07 |
| 01.13.2023 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 06.15.2023 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |