Department of Mathematical Sciences

Mathematics

WISKUNDE 365:  ANALYSIS II

Introductory Topology and Real Analysis

The topological concepts that are covered in the first part of the module continue with and extend those handled in Mathematics 324. Principally studying metric spaces, we will investigate different notions of continuity, sequences and convergence as well as compactness, connectedness and completeness. Our study of metric spaces will lead us to investigate some introductory topology – a further generalisation of metric space theory. The latter part of the course will cover more recent applications of this metric and topological theory.

Module specifications

  • 21539 365 (16) Mathematics 365
  • Year 3, Semester 2 of the Programme in Mathematical Sciences.
  • Lecturing load: 3 lectures and one turorial of 3 hours per week
  • Prerequisite modules: Prerequisite pass (PP³50) Mathematics 214, 244; Prerequisite (CM³40) Mathematics 324
  • Language specification : A

Lecturer

Learning material

Notes will be provided.

Learning Opportunities

The course material is completely covered during lectures. Tutorials provide the opportunity to solve problems under supervision and clarify uncertainties relating to the course material. Solutions of the tutorial problems are available one week after the tutorial.

Assessment

  • Method: Obtain class mark (KP=40) and pass the examination (PP=50).
  • The class mark is determined by occassional short tests at the end of tutorial periods and/or by exercises that are given during the semester (30%), together with the official class test (70%). The class test will be graded and returned as soon as possible.
  • Formula for your final mark: PP = 0,4 KP + 0,6 EP, where EP is the examination mark.
  • The dates and times of the class test and exams are announced by the University’s EXAMINATION AND TEST TIMETABLE guide. Consult the University prospectus (year book) parts 1 and 5 for further details concerning examination regulations.
  • Exercises are assessed, and the class test marked as quickly as possible and returned. Final marks are announced on the date as determined by the University’s official year plan. Students may discuss their examination scripts with the lecturer, but this may only take place after the final day for submission of marks (consult the year plan), and must be within two months of the examination being written.

General Information
Attendance of all tutorials is compulsory, also for students who are repeating the course.

Rationale
The module is offered within the Programme for Mathematical Sciences and is one of the core modules required to obtain the B.Sc. degree with Mathematics as major subject.

Outcomes

A student who has passed this module, should have the following knowledge and skills:

  • Know and understand the basic concepts and theorems in topological and metric spaces.
  • Understand the role and importance of, and interaction between, the Bolzano-Weierstrass and Heine-Borel theorems, Cantor’s “cutting” theorem and Lindelöf’s covering theorem.
  • Understand concepts allied to compactness in metric spaces, differentiate between various compactness notions and work with different convergence concepts for sequences of functions.