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 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 abstrac 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 Facult (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
60 further hours' study
For cases of fraudulent performance of exercises or tests, the provisions of the "Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións"
Contingency plan
Scenario 2.
The expository teaching will be carried out completely virtually in its scheduled schedule. They will be recorded with the media of the center. The recordings will be protected by intellectual property, will be for the exclusive use of students enrolled in the course and may not be copied. Continuous assessment will account for 30% of the final grade. The final exam will be held in person and will represent 70% of the final grade.
Scenario 3.
The expository teaching will be carried out completely virtually in its scheduled schedule. They will be recorded with the media of the center. The recordings will be protected by intellectual property, will be for the exclusive use of students enrolled in the course and may not be copied. The “interactive laboratory classes” will be replaced by tutorials for solving the proposed problems using the Teams application at the corresponding time. The continuous assessment will account for 30% of the final grade. The final exam will be held in person and will represent 70% of the final grade.
Enrique Macías Virgós
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813153
- quique.macias [at] usc.es
- Category
- Professor: University Professor
Antonio M. Gómez Tato
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813151
- antonio.gomez.tato [at] usc.es
- Category
- Professor: University Lecturer
Fernando Alcalde Cuesta
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813142
- fernando.alcalde [at] usc.es
- Category
- Professor: University Lecturer
David Mosquera Lois
- Department
- Mathematics
- Area
- Geometry and Topology
- david.mosquera.lois [at] usc.es
- Category
- Ministry Pre-doctoral Contract
Álvaro Carballido Costas
- Department
- Mathematics
- Area
- Geometry and Topology
- alvaro.carballido.costas [at] usc.es
- Category
- Xunta 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 |
| Friday | |||
| 09:00-10:00 | Grupo /CLIL_06 | Galician, Spanish | Classroom 07 |
| 09:00-10:00 | Grupo /CLIL_05 | Spanish, Galician | Classroom 08 |
| 09:00-10:00 | Grupo /CLIL_02 | Spanish | Classroom 09 |
| 10:00-11:00 | Grupo /CLIL_08 | Galician, Spanish | Classroom 07 |
| 10:00-11:00 | Grupo /CLIL_07 | Galician, Spanish | Classroom 08 |
| 01.10.2022 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 07.01.2022 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |