Prerequisite Courses:
Course Language:
English
Course Objectives:
To develop the necessary background for modern analysis courses to follow
Course Content:
Basic concepts about topological spaces and metric spaces. Complete metric spaces, Baire’s theorem, Contracting mapping theorem and its applications. Compact spaces, Arzela-Ascoli Theorem Seperability, second countability, Urysohn's lemma and the Tietze extension theorem, Connected spaces, Weierstrass approximation theorem.
Course Methodology:
1: Lecture, 2: Problem Solving
Course Evaluation Methods:
A: Written examination, B: Homework
Vertical Tabs
Course Learning Outcomes
| Learning Outcomes | Program Learning Outcomes | Teaching Methods | Assessment Methods |
| 1) Learns basic concepts of topological spaces with emphasis on metric spaces | 1,2,3,4 | 1,2 | A |
| 2) Learns Cauchy sequences and completeness | 1,2,3,4 | 2,2 | A |
| 3) Learns the concept of compact space | 1,2,3,4 | 1,2 | A |
| 4) Learns Baier’s category | 1,2,3,4 | 1,2 | A |
| 5) Learns Ascoli-Arzela theorem, Weierstrass approximation | 1,2,3,4 | 1,2 | A |
| 6) Acquires the skill of applying these concepts | 1,2,3,4 | 1,2 | A |
Course Flow
| Week | Topics | Study Materials |
| 1 | Basic concepts about metric spaces and examples | |
| 2 | Open, closed sets, topology and convergence | |
| 3 | Cauchy sequences and complete metric spaces, Baire's theorem | |
| 4 | Continuity and uniformly continuity, spaces of continuous functions, Euclidean space | |
| 5 | Contracting mapping theorem and its applications | |
| 6 | The definition of topological spaces and some examples , elementary concepts, Open bases and open subbases | |
| 7 | Compact spaces, Products of spaces,Tychonoff's theorem and locally compact spaces | |
| 8 | Compactness for metric spaces | |
| 9 | Arzela-Ascoli Theorem | |
| 10 | Seperability, second countability | |
| 11 | Hausdorff spaces, Completely regular spaces and normal spaces | |
| 12 | Urysohn's lemma and the Tietze extension theorem | |
| 13 | Connected spaces, The components of a space, Totally disconnected spaces, Locally connected spaces | |
| 14 | The Weierstrass approximation theorem ,The Stone-Weierstrass theorems |
Recommended Sources
| Textbook |
1. S. Kumaresan, Topology of Metric Spaces
2. George F. Simmons, Topology and Modern Analysis 3. W A Sutherland, Introduction to Metric and Topological Spaces 4. E T Copson, Metric Spaces |
| Additional Resources |
Material Sharing
| Documents | |
| Assignments | |
| Exams |
Assessment
| IN-TERM STUDIES | NUMBER | PERCENTAGE |
| Mid-terms | 2 | 100 |
| Quizzes | - | 0 |
| Assignments | - | 0 |
| Total | 100 | |
| CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE | 60 | |
| CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE | 40 | |
| Total | 100 |
| COURSE CATEGORY | Expertise/Field Courses |
Course’s Contribution to Program
| No | Program Learning Outcomes | Contribution | ||||
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry) | X | ||||
| 2 | Acquiring fundamental knowledge on fundamental research fields in mathematics | X | ||||
| 3 | Ability form and interpret the relations between research topics in mathematics | X | ||||
| 4 | Ability to define, formulate and solve mathematical problems | X | ||||
| 5 | Consciousness of professional ethics and responsibilty | X | ||||
| 6 | Ability to communicate actively | X | ||||
| 7 | Ability of self-development in fields of interest | X | ||||
| 8 | Ability to learn, choose and use necessary information technologies | X | ||||
| 9 | Lifelong education | X | ||||
ECTS
| ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION | |||
| Activities | Quantity |
Duration (Hour) |
Total Workload (Hour) |
| Course Duration (14x Total course hours) | 14 | 4 | 56 |
| Hours for off-the-classroom study (Pre-study, practice) | 14 | 5 | 70 |
| Mid-terms (Including self study) | 2 | 15 | 30 |
| Quizzes | - | ||
| Assignments | - | ||
| Final examination (Including self study) | 1 | 19 | 175 |
| Total Work Load | 175 | ||
| Total Work Load / 25 (h) | 7 | ||
| ECTS Credit of the Course | 7 | ||