Topology of Euclidean spaces

The study of the real line and continuous real-valued maps weas treated in the course “Introduction to mathematical analysis”. The main purpose of this course is to address the investigation of the topology of Euclidean spaces of any dimension. More specifically:

- Studying concepts, methods, and properties of metric spaces, with particular emphasis on the n-dimensional Euclidean space.

- Applying techniques related to the convergence of sequences to the study of properties associated with topology. Studying the notion of completeness.

- Studying continuous maps between metric spaces, focusing on these between Euclidean spaces. Providing examples of maps that illustrate different properties or serve to define subsets of Euclidean space. Expressing simple geometric transformations analytically.

- Studying the concepts of connectedness and compactness. Understand how these concepts allow us to generalize the fact that continuous maps defined on a closed and bounded interval and taking real values reach a maximum and a minimum, as well as all intermediate values between them.

Topic 1. Metric and Euclidean spaces (4 lecture hours)

- Metric and topological space. Vector space with inner product. Normed space. Euclidean space.
- Cauchy–Schwarz inequality and Minkowski inequality.
- Open and closed balls.
- Distance between sets. Bounded sets. Diameter.


Topic 2. The topology of metric and Euclidean spaces (4 lecture hours)

- Open and closed sets.
- The topology of metric and Euclidean spaces.
- Spaces and subspaces.
- Relative topology.


Topic 3. Convergence and completeness (4 lecture hours)

- Sequences and convergence. Subsequences.
- Convergence and topology.
- Cauchy sequences.
- Completeness.
- Completeness of Euclidean spaces.


Topic 4. Continuity (8 lecture hours)

- Continuity at a point. Global continuity.
- Global characterizations of continuity.
- Sequential continuity.
- Combined maps.
- Homeomorphisms. Topological properties.


Topic 5. Connectedness (4 lecture hours)

- Separation. Connected sets.
- Connectedness and continuity.
- Path-connected sets.


Topic 6. Compactness (4 lecture hours)

- Open cover. Compact sets.
- Compactness and continuity.
- Heine–Borel Theorem.

Basic bibliography:

- Course at Virtual Campus, available at http://xtsunxet.usc.es/carlos/topoloxia1/.
- LIMA, E.L. Espaços métricos. Brasilía: Instituto de Matemática Pura e Aplicada. Projeto Euclides (IMPA), 1993.
- MASA VÁZQUEZ, X.M. Curso de topoloxía: dos números reais ao Grupo de Poincaré. USC Editora. Manuais, Universidade de Santiago de Compostela, 2020. (Edición revisada y actualizada del manual de 1999).


Complementary bibliography:
- BARTLE, R.G. Introducción al Análisis Matemático. Ed. Limusa. México, 1980.
- BUSKES, G., VAN ROOIJ, A. Topological spaces. Springer, 1996.
- CHINN, W.G., STEENROOD, N.E. Primeros conceptos de Topología. Ed. Alhambra, 1975.
- SUTHERLAND, W.A. Introduction to metrics and topological spaces. Clarendon Press. Oxford, 1975.

The learning outcomes of this course are:

Knowledge:

Con01: Knowing the key concepts, methods, applications, and results of the various branches of Mathematics.
Con02: Understanding and using mathematical language to construct and comprehend proofs and to formulate mathematical models.
Con03: Knowing the proofs of relevant theorems in different areas of Mathematics.
Con04: Assimilating definitions of mathematical objects, relating them to others, and being able to use them in different contexts.
Con05: Abstracting essential properties and facts from a problem and identifying the appropriate mathematical tools to address it.

Skills or Abilities:

H/D01: Applying both theoretical and practical knowledge, as well as analytical and abstract thinking, to define and formulate problems and to search for their solutions in academic and professional contexts.
H/D02: Using general and subject-specific mathematical bibliography and research tools.
H/D03: Organizing and planning work effectively.
H/D04: Verifying or checking arguments and conjectures, identifying errors, and proposing revisions or counterexamples.
H/D06: Reading scientific texts both in the native language and in other relevant languages in the scientific field.
H/D07: Constructing mathematical proofs, formulating conjectures, and designing strategies to confirm or refute them.
H/D08: Proposing, analyzing, validating, and interpreting models of real-world situations using the most suitable mathematical tools for the intended objectives.

Competencies:

Comp01: Gathering and interpreting data, information, and relevant results, drawing conclusions, and writing well-reasoned reports in scientific, technological, or other contexts requiring mathematical tools.
Comp02: Communicating mathematical knowledge, procedures, results, and ideas, both in writing and orally, to both specialized and general audiences.
Comp03: Studying and learning independently new knowledge and techniques from different branches of Mathematics.
Comp04: Designing and developing algorithms and mathematical methods to solve problems in any field.

Theoretical classes will mainly involve instructor-led lessons focused on presenting theoretical content and solving selected problems or exercises. At times, this approach will resemble a traditional lecture format, while in other cases, greater student engagement will be encouraged.

Interactive sessions will aim, in some cases, at developing practical skills, and in others, at providing immediate illustration of theoretical-practical content through problem-solving, theoretical applications, exercises, or programming tasks.

Tutorial sessions will allow students to discuss specific questions related to assigned tasks or clarify doubts regarding the subject matter.

Virtual campus will provide access to all theoretical aspects of the course. Periodically, exercise sheets and questions will be distributed to students through this virtual platform.

In accordance with the Verification Report of the Degree in Mathematics, evaluation must serve to verify that students have assimilated the basic knowledge transmitted to them and have acquired the knowledge, skills, and competencies associated with the degree.

Two types of assessment are considered: continuous evaluation and final evalution. On the one hand, continuous evaluation will consist of one or two written tests administered during class hours throughout the semester. The grade obtained through continuous assessment will be applicable to both of the academic year’s examination periods (second semester and July). On the other hand, final evaluation will consist of a written exam held on the date designated by the Faculty of Mathematics. If a student does not attend the scheduled final exam in either of the two examination periods, they will receive a grade of “Absent”, even if they have participated in the continuous assessment.

Both the continuous assessment tests and the final exam will include the following components, with the corresponding approximetely weight range (in parentheses):
- Written response to theoretical questions, statement of theorems and development of proofs (around 50%).
- Written responses to practical questions, problem/exercise solving (around 50%).

Again, in accordance with the Verification Report of the Bachelor's Degree in Mathematics, the final grade for each student “may not be lower than the grade obtained in the final evaluation or the weighted average of the final and continuous evaluation”. Therefore, the final grade (FG) will be calculated using the continuous assessment grade (CA) and the final assessment grade (FA) as follows: FG = max{0.3×CA + 0.7×FA, FA}.

Tests will be equivalent in all groups of the course, although not necessarily identical.

Theoretical Teaching: 28 hours
Interactive Teaching (Laboratory/Computer Room): 28 hours
Tutorials in very small groups: 2 hours
Student's Personal Work: 92 hours
Total: 150 hours

A significant amount of time in the course is dedicated to solving exercises. Obviously, this is considered a fundamental aspect of learning the subject. However, this should not lead to the assumption that theory is less important: on the contrary, theory is the keystone of the education. In this regard, the proofs of the results not only help in understanding them better but also allow students to become familiar with the most important techniques of the course.