Learning outcomes
General topology is roughly speaking "the geometry of analysis", allowing to adapt the notions of continuity and convergence to the needs of a mathematical problem.
This course is an introduction to the basic notions of topology, which are essential to be able to attend a course of functional analysis.
Content
The course has seven chapters:
Fundamental definitions and properties (open and closed sets, neighborhoods, convergent sequences).
Bases, sub-bases and local bases (fundamental neighborhoods)(fundaments of a topology).
Continuous Functions (global, local, and sequential continuity) and equivalent topological spaces.
Countability (working with sequences) and separation (unique limits).
Compactness (open cover and extraction of a finite subcover) (Alexander's theorem and Heine-Borel theorem, properties, compactness in R^{n}) and sequential compactness (extraction of a convergent subsequence).
Product spaces (product topology, Tychonoff's product compactness theorem).
Metric spaces (metric topology, compactness in metric spaces (compactness equivalent to sequential compactness), complete metric spaces (compact = closed and totally bounded).
Recommended readings
- Adams, Colin et Robert Franzosa, Introduction to topology : pure and applied, Pearson Prentice Hall, Upper Saddle River, NJ, 2008. [BUMP: 2710]
- Lipschutz, Seymour, Topologie - Cours et problemes, Serie Schaum, McGraw-Hill, New York 1965, Paris (pour la traduction francaise) 1981. [BUMP : SB08413/006]
Language of instruction
French / Français
Location for course
NAMUR
Organizer
Faculté des sciences
Rue de Bruxelles, 61
5000 NAMUR
Degree of Reference
Undergraduate Degree