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	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

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
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
#	Program Qualifications / Competencies	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)
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