# Topology (MATH372) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Topology MATH372 Area Elective 3 0 0 3 6
Pre-requisite Course(s)
MATH 251
Course Language English Elective Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Discussion, Question and Answer, Team/Group, Brain Storming. The aim of this course is to introduce the student some algebraic and differential topological ideas at an early stage emphasizing unity with geometry and more generally introduce the student to the relation of the modern axiomatic approach in mathematics to geometric intuition. The students who succeeded in this course; prove elementary theorems involving sets and functions, understand topological spaces and continuous functions, determine whether a topological space has any of various topological properties, 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, understand the classification of surfaces. Fundamental concepts, functions, relations, sets and Axiom of Choice, well-ordered sets, topological spaces, basis, the Order Topology, the Subspace Topology, closed sets and limit points, continuous functions, the Product Topology, Metric Topology, the Quotient Topology, connectedness and compactness, Countability and Separation Axioms, the fundam

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Fundamental Concepts, Functions pp. 4-20
2 Relations, Sets and Axiom of Choice, Well-ordered Sets pp. 21-50, 57-66
3 Topological Spaces, Basis, The Order Topology pp. 75-86
4 The Subspace Topology, Closed Sets and Limit Points pp. 88-100
5 Continuous Functions, The Product Topology pp. 102-117
6 Metric Topology, The Quotient Topology pp. 119-126, 136-144
7 Midterm
8 Connectedness pp. 147-162
9 Compactness pp. 163-185
10 Countability and Separation Axioms pp. 190-222
11 Homotopy, The Fundamental Group pp.322-334
12 Covering Spaces, The Fundamental Group of The Circle, Retractions and Fixed Points pp. 335-353
13 The Fundamental Theorem of Algebra, The Borsuk-Ulam Theorem, Homotopy Type pp. 353-365
14 Fundamental Groups of Surfaces pp. 368-375
15 The Classification Theorem pp. 462-476
16 Final Exam

### Sources

Course Book 1. J.R. Munkres, Topology, Second Edition, Prentice Hall, NJ, 2000. 2. M. C. Gemignani, Elementary Topology, Addison-Wesley, 1972 3. M. D. Crossley, Essential Topology, Springer-Verlag, 2005 4. L. C. Kinsey, Topology of Surfaces, Springer-Verlag, 1997

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 3 15
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 50
Final Exam/Final Jury 1 35
Toplam 6 100
 Percentage of Semester Work 65 35 100

### Course Category

Core Courses X

### 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 X
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 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 X
4 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 X
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 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 X
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. X
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. X

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 3 16 48
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 12 24
Prepration of Final Exams/Final Jury 1 18 18