ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician, English
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Center Faculty of Mathematics
Call: Second Semester
Teaching: Sin Docencia (En Extinción)
Enrolment: No Matriculable (Sólo Planes en Extinción)
The study of the topology of the real line began in the subject "Introduction to Mathematical Analysis" and, in reference to continuity, it is developed in the subject "Continuity and derivability of functions of a real variable". Now, in this subject, the study of topology is addressed not only of the real line, but also of Euclidean spaces of any dimension. In addition, a more systematic treatment of the issues considered will be made.
The main objectives are:
• Study concepts, methods and metric and, fundamentally, topological properties in R^n, starting from its Euclidean structure.
• Apply sequence convergence techniques to the study of properties related to topology. Study completeness.
• Study the continuity of functions in the field of Euclidean spaces. Identify continuous functions, or discontinuities of functions. Describe functions geometrically. Give examples of functions that illustrate various properties. Express simple geometric transformations analytically.
• Understand the concepts of connectedness and compactness. In its simplest expression, a typical result will say that every real function continuous with domain a closed interval reaches the maximum, the minimum and any intermediate value; it will be noted that the only necessary properties of the interval are connectedness and compactness. It is a sample of one of the most characteristic aspects of mathematics: how the solution of problems, sometimes of simple formulation, often requires very abstract theories.
Topic 1 Euclidean spaces (4 expositive hours)
1.1 Scalar product and Euclidean norm
1.2 Cauchy-Schwarz and Minkowski inequalities
1.3 Euclidean distance. Properties; triangle inequality
1.4 Open Balls
1.5 Distance between sets. Bounded sets. Diameter
Topic 2 The topology of Euclidean space (4 expositive hours)
2.1 Open set definition
2.2 Characteristic properties of open sets
2.3 Closed sets
2.4 Spaces and subspaces. relative open
Topic 3 Convergence and completeness (4 expository hours)
3.1 Successions. convergent sequences. Subsequences.
3.2 Convergence and topology
3.3 Cauchy sequences
3.4 Completeness of Euclidean space
Topic 4 Continuity (8 expositive hours)
4.1 Definition of continuity
4.2 Global characterizations of continuity
4.3 Sequential continuity
4.4 Combined function
4.5 Homeomorphisms
4.6 Topological properties
Topic 5 Connection (4 expository hours)
5.1 Connected sets
5.2 Connection and continuity
5.3 Path-connected sets
Topic 6 Compactness (4 expositive hours)
6.1 Compactness
6.2 Compactness and continuity
6.3 Characterization of compact sets in Euclidean space (Heine-Borel Theorem)
Basic bibliography:
Course on the virtual campus, also accessible at http://xtsunxet.usc.es/carlos/topoloxia1/
MASA VAZQUEZ, X.M. Topology course: two real numbers to the Poincaré Group. USC Publisher. Manuais, Universidade de Santiago de Compostela, 2020. (Revised and updated edition of the 1999 manual)
MASA VAZQUEZ, X.M. Xeral topology. Introduction to Euclidean, metric and topological spaces. University Manuals, University of Santiago de Compostela, 1999.
Complementary bibliography:
BARTLE, R.G. Introduction to Mathematical Analysis. Ed. Limusa. Mexico, 1980.
BUSKES, G. AND VAN ROOIJ, A. Topological spaces. Springer, 1996.
https://link.springer.com/book/10.1007/978-1-4612-0665-1
CHINN, W.G. and STEENROOD, N.E. First concepts of Topology. Ed. Alhambra, 1975.
SUTHERLAND, W.A. Introduction to metrics and topological spaces. Clarendon Press. Oxford, 1975.
This course aims to contribute to improving the basic, general and transversal skills of the Mathematics Degree. In addition, the following SPECIFIC competences of the degree will be worked on:
CE1 - Understand and use mathematical language.
CE2 - Know rigorous proofs of some classical theorems in different areas of Mathematics.
CE3 - Devise proofs of mathematical results, formulate conjectures and imagine strategies to confirm or deny them.
CE4 - Identify errors in incorrect reasoning proposing demonstrations or counterexamples.
CE5 - Assimilate the definition of a new mathematical object, relate it to others already known, and be able to use it in different contexts.
CE6 - Know how to abstract the properties and substantial facts of a problem, distinguishing them from those that are purely occasional or circumstantial.
The assessment will consist on an exam whose date will be fixed by the Faculty of Mathematics.
Victor Sanmartin Lopez
- Department
- Mathematics
- Area
- Geometry and Topology
- victor.sanmartin [at] usc.es
- Category
- PROFESOR/A PERMANENTE LABORAL
| 05.21.2026 10:00-14:00 | Grupo de examen | Classroom 06 |
| 07.08.2026 10:00-14:00 | Grupo de examen | Classroom 06 |