6 ECTS credits
150 h study time

Offer 1 with catalog number 1012235BNR for all students in the 2nd semester at a (B) Bachelor - advanced level.

Semester
2nd semester
Enrollment based on exam contract
Impossible
Grading method
Grading (scale from 0 to 20)
Can retake in second session
Yes
Enrollment Requirements
Students must have followed ‘Analysis and Topology in Metric Spaces’, before they can enroll for ‘Topology’.
Taught in
Dutch
Faculty
Faculty of Science and Bio-engineering Sciences
Department
Mathematics
Educational team
Mark Sioen (course titular)
Activities and contact hours
26 contact hours Lecture
13 contact hours Seminar, Exercises or Practicals
Course Content
In Analysis many examples of metric spaces are encountered. Starting from Euclidean spaces a lot of other situations arise naturally where distances are measured in a canonical way. For example subspaces of Euclidean spaces, or Hilbert spaces and all kinds of function spaces carry natural metrics.

Topologists make a classification of such spaces. Spaces are said to be equivalent if they are homeomorphic. This means that one space can be continuously transformed into the other. The idea is that of a rubber sheet geometry, that is to say we do not mind stretching our space. Distances do not really matter. For example the real line is homeomorphic to the open unit interval. In some cases it is not easy to construct the homeomorphism explicitly. It is even more difficult to show that such a homeomorphism does not exist. That is why topological invariants are introduced. These invariants are preserved under homeomorphisms. For instance, since the real line is not compact, it cannot be homeomorphic to the closed unit interval. Compactness is an invariant. Other invariants that will be studied are separation properties, connectedness and separability.
In making the classification mentioned above, certain constructions play an important role. Products and quotients are such fundamental constructions.
We will see that these constructions are not always meaningful in the setting of metric spaces. This fact has important consequences: for instance pointwise convergence of sequences of functions cannot generally be described by a metric.
In this course we build a new framework for such situations. Abstract Topological spaces will be introduced by axiomatizing the notion of open set.
Continuity and convergence will be described in this new setting and we will prove that the fundamental constructions mentioned above, can always be carried out in the topological framework. Topological invariants will be generalized to the new setting.
Finally the abstract theory will be applied in order to characterize compact subspaces of function  spaces. So the course ends where Topology originally started. Indeed the problems encountered in the study of function spaces have contributed to the birth of Topology in the beginning of the century.
Course material
Course text (Required) : Topologie, Beknopte syllabus is beschikbaar over alle hoofdstukken (behalve voor het hoofdstuk functieruimten)., Colebunders, VUB, 2220170000428, 2015
Handbook (Recommended) : Topology, K. Janich, Springer Verlag, 9781461270188, 2012
Handbook (Recommended) : General Topology, S. Willard, Addison and Wesley, 9780486434797, 2004
Handbook (Recommended) : Topology for Analysis, A. Wilansky, Ginn & Co, 9780486469034, 2008
Handbook (Recommended) : General Topology, J. L. Kelley, Van Nostrand, 9780486815442, 2017
Handbook (Recommended) : Topologie Générale, chap.I, Chapitres 1, N. Bourbaki, Hermann, 9783540339366, 2007
Additional info

Syllabus is available

 

Learning Outcomes

General competencies

- The student can make abstraction of the metric context and masters notions of convergence and continuity in a topological setting.

- The student has insight in the essential differences compared to the metrizable context and knows their impact on the study of compactness.

- The can use topological properties to classify spaces.

- The student is used to work with initial and final constructions, recognizes the importance with respect to the study of function spaces and has insight in results such as the theorem of Ascoli-Arzela for uniform convergence on compact subsets.

 

Construction & formulation

- The student can analyse proofs and understands the logical reasoning behind them. For a given proposition the student is aware of the role the conditions play in the proof.

 

- The student can complete easy proofs that are left as an exercise or that are only partially explained in the syllabus or in class. The missing arguments can be filled in independently.

 

- The student masters the mathematical language and is able to produce correct mathematical formulations and proofs.

Insight and connections

- The student has an overall insight in the material, has a deep understanding of new concepts and results and is aware of the connection between de various concepts.

- The student is able to make the link between concepts on one hand and illustrating examples on the other hand.

 

- The student has insight in the relation to analogous concepts as they were introduced in other courses.

Independent problemsolving

- The student can independently solve problems: he/she is able to recognize a problem, to choose an appropriate strategy, to select the most suitable method

Grading

The final grade is composed based on the following categories:
Oral Exam determines 60% of the final mark.
Written Exam determines 40% of the final mark.

Within the Oral Exam category, the following assignments need to be completed:

  • Mondeling theorie examen with a relative weight of 1 which comprises 60% of the final mark.

Within the Written Exam category, the following assignments need to be completed:

  • Schriftelijk oefeningen examen with a relative weight of 1 which comprises 40% of the final mark.

Additional info regarding evaluation
Oral examination (theoretical part) 60%
Written examination (exercises) 40%
Allowed unsatisfactory mark
The supplementary Teaching and Examination Regulations of your faculty stipulate whether an allowed unsatisfactory mark for this programme unit is permitted.

Academic context

This offer is part of the following study plans:
Bachelor of Mathematics and Data Science: Standaard traject (only offered in Dutch)