ECTS - Topology
Topology (MATH571) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Topology | MATH571 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Ph.D. |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Discussion, Question and Answer. |
| Course Lecturer(s) |
|
| Course Objectives | This course is designed to provide necessary background and further knowledge in Topology for graduate students of Mathematics. The content of the course serves to lay the foundations for future study in analysis, in geometry, and in algebraic and geometric topology. |
| Course Learning Outcomes |
The students who succeeded in this course;
|
| Course Content | Topological spaces, homeomorphisms and homotopy, product and quotient topologies, separation axioms, compactness, connectedness, metric spaces and metrizability, covering spaces, fundamental groups, the Euler characteristic, classification of surfaces, homology of surfaces, simple applications to geometry and analysis. |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | Metric Spaces, Topological Spaces, Subspaces, Connectivity and Components, Compactness | pp. 1-14, 18-22 |
| 2 | Products, Metric Spaces Again, Existence of Real Valued Functions, Locally Compact Spaces, Paracompact Spaces | pp. 22-39 |
| 3 | Quotient spaces, homotopy, Homotopy Groups | pp. 39-51, 127-132 |
| 4 | The Fundamental Group, Covering Spaces | pp. 132-143 |
| 5 | The Lifting Theorem, Deck Transformations | pp. 143-150 |
| 6 | Properly Discontinuous Actions, Classification of Covering Spaces, The Seifert-Van Kampen Theorem | pp. 150-164 |
| 7 | Homology Groups, The Zeroth Homology Group, The First Homology Group | pp. 168-175 |
| 8 | Functorial Properties, Homological Algebra, Computation of Degrees | pp. 175-194 |
| 9 | Midterm | |
| 10 | CW-Complexes, Cellular Homology | pp. 194-207 |
| 11 | Cellular Maps, Euler’ s Formula, Singular Homology | pp. 207-211, 215-217, 219-220 |
| 12 | The Cross Product, Subdivision, The Mayer-Vietoris Sequence | pp. 220-230 |
| 13 | The Borsuk-Ulam Theorem, Simplicial Complexes | pp. 240-250 |
| 14 | Simplicial Maps | pp. 250-253 |
| 15 | The Lefschetz-Hopf Fixed Point Theorem | pp. 253-259 |
| 16 | Final Exam |
Sources
| Course Book | 1. Glen E. Bredon, Topology and Geometry, Springer-Verlag, NY, 1993. |
|---|---|
| Other Sources | 2. J.R. Munkres, Topology, Second Edition, Prentice Hall, NJ, 2000. |
| 3. A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 5 | 30 |
| Presentation | - | - |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 30 |
| Final Exam/Final Jury | 1 | 40 |
| Toplam | 7 | 100 |
| Percentage of Semester Work | 60 |
|---|---|
| Percentage of Final Work | 40 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Is independently able to build a problem in the area of study, solve the problem by developing solution techniques and assess the solutions. | X | ||||
| 2 | Is capable of creating a groundwork in the fundamental branches of mathematics as well as in his/her research area | X | ||||
| 3 | follows the latest national and international literature in Mathematics and in his/her area of research; and uses them in his/her related studies | X | ||||
| 4 | observes and adopts the scientific ethical values in his/her professional and social life | X | ||||
| 5 | presents in Turkish and English in academic/scientific events the results of his/her research or the latest studies and findings on a special topic and participates in discussions | X | ||||
| 6 | Develops skills to work independently or as a member of a team | X | ||||
| 7 | Develops competences in the areas of creative and critical thinking, problem solving and producing original studies. Follows recent scientific studies, is capable of making an analysis, synthesis and assessment of the knowledge acquired | X | ||||
| 8 | Is open to lifelong improvement of his/her acquired knowledge, skills and competences. | X | ||||
| 9 | Is able to apply the acquired knowledge and problem-solving skills to interdisciplinary studies, proposes different solution methods to problems in terms of mathematical models and from a mathematical point of view | X | ||||
| 10 | Uses the mathematical based softwares, informatics and communication technologies for scientific purposes | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | |||
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 3 | 42 |
| Presentation/Seminar Prepration | |||
| Project | |||
| Report | |||
| Homework Assignments | 5 | 3 | 15 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 10 | 10 |
| Prepration of Final Exams/Final Jury | 1 | 10 | 10 |
| Total Workload | 77 | ||