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)