6 ECTS credits
150 h study time

Offer 1 with catalog number 4013331FNR for all students in the 2nd semester at a (F) Master - specialised level.

Semester
2nd semester
Enrollment based on exam contract
Impossible
Grading method
Grading (scale from 0 to 20)
Can retake in second session
Yes
Taught in
Dutch
Partnership Agreement
Under interuniversity agreement for degree program
Faculty
Faculty of Science and Bio-engineering Sciences
Department
Mathematics
External partners
Universiteit Antwerpen
Educational team
Robert LOWEN (course titular)
Eva Colebunders
Anna-Karina Segers
Activities and contact hours
30 contact hours Lecture
30 contact hours Seminar, Exercises or Practicals
Course Content
We develop a setting to deal with non-topological concepts of classical analysis, such as completeness, total boundedness, uniform convergence and uniform continuity.  The new setting is the one of uniform spaces and is significantly wider than the context of metric spaces, including for example topological groups, topological vectorspaces and several natural function spaces. Uniform spaces share an important quality of metric spaces namely that 'distance' is a notion which can be applied uniformly to pairs of points without regard to their location.
In the setting of uniform spaces we develop the theory of completion and we relate the results to constructions of compactifications.
In this context an important role is played by the ring of realvalued continuous maps on a topological space.  Extension problems with regard to continuous functions are studied.

1.  Uniform spaces and uniformly continuous maps
  -  Families of pseudometrics
  -  Topological groups
  2.  Uniformizability
  -  Compact Hausdorff spaces
  -  Complete regularity
  3.  Completeness and Total boundedness
  -  Completions
  -  Compactifications
  4. Extensions of continuous maps
  - Normality
  -Tietze’s theorem
Course material
Digital course material (Required) : Gevorderde topologie, Cursusnota’s en oefeningen worden door de docent verstrekt, UA/VUB
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.II, N. Bourbaki, Hermann, 9783540339366, 2007
Additional info
Handouts

Additional Study Material
N. Bourbaki, Topologie  Générale   chap II, Hermann
   L. Gillman & M. Jerison, Rings of continuous functions, Springer Verlag
   J. Dugundji, Topology, Allyn & Bacon
   J. L. Kelley, General Topology, Van Nostrand
   A. Wilansky, Topology for Analysis, Ginn & Co
   H.Herrlich, Uniforme Raüme, Heldermann
S. Willard, General Topology, Addison and Wesley
Learning Outcomes

General competencies

Link to UA

Grading

The final grade is composed based on the following categories:
Other Exam determines 100% of the final mark.

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

  • Mondeling examen with a relative weight of 1 which comprises 100% of the final mark.

Additional info regarding evaluation
Oral exam
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:
Master of Mathematics: Fundamental Mathematics (only offered in Dutch)
Master of Mathematics: Education (only offered in Dutch)