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Topology

Code: 100106
Credits: 6
2026/2027
Degree programme Type Course
Mathematics OB 2

Contact lecturer

Name :
Carlos Broto Blanco
Email :
carles.broto@uab.cat

Teaching staff

Michael Janou Glaeser Santamaría

Group languages

You can consult this information at the end of the document.

Prerequisites

It is extraordinarily important that students have assimilated, before starting the course, the basic foundations of deductive mathematical reasoning. They must have experience with the axiomatic method, must know the most basic principles of mathematical logic, and should be familiar with what constitutes correct mathematical reasoning versus incorrect reasoning, as well as with the various paradigms of mathematical proof. They must be skilled in the negation of a proposition, in the use of quantifiers, and in the concept of implication. Mastery of the concept of continuity from calculus courses is also important.


Objectives

The main objective of the course is to understand, internalize and master the concept of topology, which is an abstraction that allows the notion of continuity to be used with great flexibility. Specifically, we introduce the concepts of topological space and continuous application, as a generalization of the notion of continuity to an environment devoid of metrics and we acquire the necessary fluency with the vocabulary and basic machinery of topology. We observe examples that appear in different areas and situations with the idea of ​​acquiring the ability to apply these ideas in new situations. The basic notions and properties of separation, connection and compactness will be developed. Finally, we will introduce the notion of topological variety and the classification of compact surfaces as a final objective.

Learning outcomes

  • CM28 (Evaluate topological and metric properties to distinguish different surfaces.) Evaluate topological and metric properties to distinguish different surfaces.
  • CM30 (Communicate advanced topology and geometry results to a specialist and non-specialist audience.) Communicate advanced topology and geometry results to a specialist and non-specialist audience.
  • KM35 (Enunciate the axioms of general topology.) Enunciate the axioms of general topology.
  • SM37 (Apply the basic concepts associated with the notions of metric space and topological space to the resolution of algebra and analysis problems.) Apply the basic concepts associated with the notions of metric space and topological space to the resolution of algebra and analysis problems.
  • SM38 (Discriminate between the intrinsic and extrinsic properties of a surface in space.) Discriminate between the intrinsic and extrinsic properties of a surface in space.

Contents

  1. Topological properties of metric spaces.
  2. Topological spaces: axioms.
  3. Topological Subspaces. Product topology. Quotient Topology.
  4. Hausdorff spaces: axioms of separability.
  5. Compact spaces.
  6. Connectedness and path connectedness.
  7. Local properties
  8. The classification of compact surfaces.

Learning activities and methodology

Title Hours ECTS Learning outcomes
Studying theoretical concepts and solving problems 85 3.4 CM28, CM30, KM35, SM37, SM38
Problem session 14 0.56 CM28, CM30, KM35, SM37, SM38
Lectures 30 1.2 CM28, CM30, KM35, SM37, SM38
Working Seminars 6 0.24 CM28, CM30, KM35, SM37, SM38

Theory classes are intended to develop the concepts, examples and results specific to the subject. Problem and seminar sessions are dedicated to solving exercises and, especially in the case of seminars, to delving into parallel or complementary issues.


It is inherent to the program and great importance will be given to the correct generation of reasoning, the detection of incorrect reasoning and the ability to communicate these reasonings using mathematical language rigorously.


It must be taken into account that the acquisition of knowledge and skills specific to the subject cannot be completed without autonomous and individual dedication to the hours reserved for this purpose.


The Virtual Campus will be used as a means of communication. The course notes, exercise lists and all teaching materials used will be published in the CV.

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Final Exam 50% 3 0.12 CM28, CM30, KM35, SM37, SM38
Mid-term exam 30% 3 0.12 CM28, CM30, KM35, SM37, SM38
Recovery exam 80% 3 0.12 CM28, CM30, KM35, SM37, SM38
Seminars 20% 6 0.24 CM28, CM30, KM35, SM37, SM38

There are two written exams: a mid-term exam (30% of the final grade) and a final exam (50% of the final grade). This is completed with a specific evaluation of the activity developed in the seminars, which will count for 20% of the final grade. Attendance at the seminars is mandatory.


A minimum of 3.5 points out of 10 must be obtained in the final exam to pass the subject. Only those who have not passed the subject with the final exam can sit the retake exam. In this exam, the partial and final exams are retaken.


If, having not achieved the minimum of 3.5 in the final exam, a person does not appear for the retake, the course grade will be the minimum between 4.5 and the weighted average of the assessment activities to which they have appeared.


Those who have not completed assessment activities that add up to a minimum of 50% of the final grade will be considered non-evaluable.


In this subject, the use of Artificial Intelligence (AI) technologies is not allowed in any of its phases.


Single assessment. Students who have opted for the single assessment modality must take a written test on the day of the final exam. They must then take an oral test of problem solving and commentary on one of the seminars chosen by the professor. In addition, they will submit written solutions of the 3 seminars at the beginning of the written test. The student's grade will be the weighted average of the two previous activities, where the theory exam will represent 80% of the grade and the oral exam 20%. If the final grade does not reach 5, the student has another opportunity to pass the subject through the retake exam that will have a weight of 80% of the final grade; the oral exam is not retaken.

Bibliography

Basic bibliography:

  • Jaume Aguadé, Apunts d'un curs de topologia elemental. http://mat.uab.es/~aguade/teaching.html
  • Czes Kosniowski, A first course in algebraic topology. Cambridge University Press 1980.

Bibliography related to topological manifolds:

  • William S. Massey, A basic course in algebraic topology. Springer-Verlag 1991.
  • John M. Lee, Introduction to topological manifolds, Graduate Text in Mathematics 202, Springer, 2011
  • Vicenç Navarro, Pere Pascual, Topologia Algebraica, Universitat de Barcelona, 1999

Bibliography:

  • James Munkres, General Topology, Prentice-Hall, 2000.
  • Rysarzd Engelking, General Topology, Sigma Series in Pure Mathematics 6, Heldermann Verlag, 1989.
  • James Dugundji, Topology, Reprinting of the 1966 original. Allyn and Bacon Series in Advanced Mathematics. Allyn and Bacon, Inc., Boston. 1978.

Free textbooks online:

  • Marta Macho-Stadler, Topología, https://www.ehu.eus/~mtwmastm/Topologia1415.pdf
  • Jesper Moller, General Topology, http://web.math.ku.dk/~moller/e03/3gt/notes/gtnotes.pdf
  • O. Viro, O Ivanov, N. Netsvetaev, V. Kharlamov, Elementary Topology Problem Textbook,
  • http://www.pdmi.ras.ru/~olegviro/topoman/eng-book-nopfs.pdf

Software

This subject does not use mandatory software..

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 1 Catalan first semester morning-mixed
(SEM) Seminars 1 Spanish first semester morning-mixed
(TE) Theory 2 Catalan second semester morning-mixed
(PAUL) Classroom practices 2 Catalan first semester morning-mixed
(SEM) Seminars 2 Spanish first semester morning-mixed
(PAUL) Classroom practices 3 Spanish second semester morning-mixed
(SEM) Seminars 3 Spanish second semester morning-mixed
(PAUL) Classroom practices 4 Spanish second semester morning-mixed
(SEM) Seminars 4 Spanish second semester morning-mixed