Topology of Euclidian Spaces

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 work in the classroom with a large group consists, fundamentally, of teaching given by the teacher. Ordinarily, part of the time will be devoted to the exposition of theoretical questions, and another part to the illustration with examples and the formulation of problems or exercises. The involvement of all students in the discussion of the issues raised will be sought.

In face-to-face teaching in small groups (seminary) theoretical-practical questions and exercises will be proposed and resolved. In teaching in smaller groups (laboratory) preference will be given to the participation of students and to clarify doubts about theory, problems and exercises.

The tutorials in very small groups will be dedicated, individually or in groups, to solving the doubts and particular difficulties that appear, and to the individualized follow-up of each student.

There will be a virtual course, where all the theoretical aspects of the subject and solved exercises are detailed. Periodically, bulletins of exercises and questions will be delivered to the student through the virtual course.

The expository and interactive teaching will be face-to-face. Tutorials can be face-to-face or done virtually. Communication with students, in addition to face-to-face, can also be done through the virtual course forums and email.

There will be a double method of evaluation: continuous evaluation, based on written tests carried out in class and participation, and punctual evaluation, through a final written test, the exam, set in the faculty calendar. The final grade will be obtained by the following formula, where AC indicates the grade of the continuous evaluation and EF that of the final exam:

max{ 0.3 AC + 0.7 EF, EF }.

The continuous evaluation will consist of one test that will be carried out in class in which students must solve the exercises that are indicated to them. Participation in lectures and interactive classes, as well as tutorials may be rewarded with up to + 1 point. The final exam will have a theory part, which may cover the definition of concepts, the statement of results or their total or partial demonstration. The other part will consist of solving exercises, which will be analogous to those proposed during the course. Each of the parts (theory-exercises) will have a weight of between 40% and 60% of the total.

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 appear for the exam set by the faculty on either of the two occasions, he/she will have the grade of “Not presented”, even if he/she has participated in the continuous evaluation.

The evaluation will be equivalent in both groups, although the exams will not be necessarily the same.

In addition to the specific skills, the general skills CG1 (Know the most important concepts, methods and results), CG3 (Apply both the theoretical and practical knowledge acquired as well as the capacity for analysis and abstraction in the definition and approach of problems and in the search for their solutions) and CG4 (Communicate -in writing--knowledge, procedures, results and ideas).

Expository classes: 28 hours
Seminar classes: 14 hours
Laboratory classes: 14 hours
Tutorials in very small groups: 2 hours
Evaluation activities: 5 hours
Non-face-to-face personal work time: 87 hours

Total: 150 hours

A lot of time is devoted to solving exercises in the course. Obviously, it is considered a fundamental aspect in learning the subject. This should not lead us to think that theory is less important: quite the contrary, theory is the cornerstone of training. It will be necessary to handle a certain number of definitions and results, which will have to be assimilated in a short period of time. The demonstrations of the results help to better understand them and allow to become familiar with the most important techniques; should constitute one of the fundamental components of the study of the subject. The other, certainly, will be the effort to solve the exercises.

In the case of fraudulent completion of exercises or tests, the provisions of the "Regulations for evaluating the academic performance of students and review of grades" will apply.