Metric Topology (MAT301) | GSÜ - ECTS Information Guide

Content

Objective

The aim of this course is to introduce the theory of metric spaces and teach the fundamental properties and applications of these structures. By learning the concept of metric spaces, students will gain a deep understanding of important results in analysis and topology. Additionally, the course aims to develop students' skills in abstract mathematical thinking.

Course Content

The course generalises to arbitrary metric spaces the notions of nearness, convergence and continuity known on R. It covers: distances, their examples and their constructions; the topological vocabulary (balls, open and closed sets, interior, closure, boundary); sequences and convergence; continuous maps and homeomorphisms; compactness; normed vector spaces; connectedness; and completeness up to the Banach fixed-point theorem.

Course Learning Outcomes

Ability to define the fundamental concepts and examples of metric spaces.
Ability to understand sequences, continuity and homeomorphisms in metric spaces.
Ability to state the different characterisations of compactness and to apply the Heine-Borel theorem.
Ability to explain connectedness and path-connectedness and to use the intermediate value theorem.
Ability to understand the continuity of linear maps and the equivalence of norms in finite dimension in normed vector spaces.
Ability to analyse completeness, the Banach fixed-point theorem and its applications.

Teaching and Learning Methods

Theoretical lectures: Fundamental concepts and theorems will be explained in class.
Applied problems: Example problems will be solved with the students to reinforce the concepts.
Quizzes and exams: Regular quizzes and a final exam will track students' progress.
Student-centered discussions: Student discussions will be encouraged through challenging problems.

References

https://github.com/onayg/mat301
An introduction to real analysis, Tosun Terzioğlu
Burroni E, La topologie des espaces métriques : niveau L3 : cours et exercices corrigés

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Preliminaries: N, Z, Q, R; the least upper bound property.
2	Distances: axioms; usual, discrete, SNCF and p-adic examples.
3	Constructions: induced distance, pullback, product, norm.
4	Balls, neighbourhoods, open and closed sets.
5	Interior, closure, boundary; topological equivalence of metrics.
6	Sequences: accumulation points, convergence, sequential characterisation of closed sets.
7	Continuous maps; Lipschitz maps; homeomorphisms.
8	Compactness: finite subcovers, Bolzano-Weierstrass.
9	Lebesgue number; Heine-Borel theorem; compact subsets.
10	Normed vector spaces: continuity of linear maps.
11	Equivalence of norms in finite dimension; Riesz's lemma; local compactness.
12	Connectedness: characterisations, connected components, intermediate value theorem.
13	Path-connectedness; convexity in normed vector spaces.
14	Completeness: Cauchy sequences, Cantor's criterion; Banach fixed-point theorem.

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	4	50
Contribution of final exam to overall grade	1	50
Total	5	100

In-Term Studies
Activities	Number	Contribution
Assignments	0	0
Presentation	0	0
Midterm Examinations (including preparation)	0	0
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	4	0
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Make-up	0	0
Total	4	0

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways;			X
2	can properly state and use concepts and results of major mathematical interest;			X
3	masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools;				X
4	is able to express one’s mathematical ideas in an organised way both in written and oral forms;			X
5	understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...);				X
6	has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems;		X
7	has a theoretical and practical knowledge in computer science well adapted for learning a programming language;	X
8	has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions;		X
9	has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement;		X
10	masters French language as well as other foreign languages, to a level sufficient to study or work abroad.		X

ECTS

Activities	Number	Period	Total Workload
Class Hours	70	1	70
Working Hours out of Class	14	4	56
Assignments	0	0	0
Presentation	0	0	0
Midterm Examinations (including preparation)	0	0	0
Project	0	0	0
Laboratory	0	0	0
Other Applications	0	0	0
Final Examinations (including preparation)	0	0	0
Quiz	6	8	48
Term Paper/ Project	0	0	0
Portfolio Study	0	0	0
Reports	0	0	0
Learning Diary	0	0	0
Thesis/ Project	0	0	0
Seminar	0	0	0
Other	0	0	0
Make-up	1	25	25
Total Workload			199
Total Workload / 25			7.96
Credits ECTS			8