Topology (MATH571) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Topology MATH571 3 0 0 3 5
Pre-requisite Course(s)
Consent of the Department
Course Language English
Course Type N/A
Course Level Natural & Applied Sciences Master's Degree
Mode of Delivery Face To Face
Learning and Teaching Strategies Lecture, Discussion, Question and Answer.
Course Coordinator
Course Lecturer(s)
Course Assistants
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;
  • prove elementary theorems involving the concepts of continuous function, compactness, connectedness, the countability and the separation axioms,
  • understand homotopy, the fundamental group of a space and covering spaces, the classification of surfaces.
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


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
Major Area Courses
Supportive Courses X
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 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

ECTS/Workload Table

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Special Course Internship
Field Work
Study Hours Out of Class 14 3 42
Presentation/Seminar Prepration
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