Mathematics

Metric Topology(MAT301)

Content
Course Code Course Name Semester Theory Practice Lab Credit ECTS
MAT301 Metric Topology 5 3 2 0 5 8
Prerequisites MAT101, MAT102
Admission Requirements MAT101, MAT102
Language of Instruction
Course Type Compulsory
Course Level Bachelor Degree
Course Instructor(s) Ayberk ZEYTİN azeytin@gsu.edu.tr (Email)
Assistant
Objective The aim of this course is to introduce the theory of metric spaces and teach the fundamental properties and applications of these structures. By learning the concept of metric spaces, students will gain a deep understanding of important results in analysis and topology. Additionally, the course aims to develop students' skills in abstract mathematical thinking.
Content This course aims to teach the fundamental concepts of the theory of metric spaces. Initially, the basic properties on R and sequences in R will be covered, followed by an introduction to the concept of metric spaces, supported by various examples. The open and closed sets in metric spaces will be discussed, and the fundamental properties of these structures will be examined. The convergence of sequences in metric spaces and the concept of complete metric spaces will be studied in detail. The concept of continuous functions and continuity in metric spaces will also be included in the course content. Furthermore, the notion of compactness in metric spaces will be emphasized and studied in detail over three weeks. In the final part of the course, the Banach fixed-point theorem and its various applications will be discussed.
Course Learning Outcomes Ability to define the fundamental concepts and examples of metric spaces.
Ability to understand the basic concepts related to sequences and functions in metric spaces.
Ability to explain the concepts of compactness, convergence, and completeness in metric spaces.
Ability to analyze the Banach fixed-point theorem and its applications.
Ability to use the concept of metric spaces to solve abstract mathematical problems.
Teaching and Learning Methods Theoretical lectures: Fundamental concepts and theorems will be explained in class.
Applied problems: Example problems will be solved with the students to reinforce the concepts.
Quizzes and exams: Regular quizzes and a final exam will track students' progress.
Student-centered discussions: Student discussions will be encouraged through challenging problems.
References An introduction to real analysis, Tosun Terzioğlu
Print the course contents
Weekly Contents
Theory Topics
Week Weekly Contents
1 Properties of the set of real numbers
2 Sequences in the set of real numbers, accumulation points of sets, limit values of sequences
3 Open and closed subsets of the set of real numbers
4 Metric spaces: definition and examples
5 Open and closed sets in metric spaces
6 Sequences and convergence in metric spaces, accumulation points of sets, limit values of sequences
7 Topological properties of metric spaces: completeness
8 Topological properties of metric spaces: compactness
9 Topological properties of metric spaces: connectedness
10 Sequences and limits in function spaces
11 Open and closed sets in function spaces
12 Topological properties of function spaces
13 Banach fixed-point theorem
14 Applications of the Banach fixed-point theorem
Practice Topics
Week Weekly Contents
Assessment System
Contribution to Overall Grade
  Number Contribution
Contribution of in-term studies to overall grade 6 60
Contribution of final exam to overall grade 1 40
Toplam 7 100
In-Term Studies
  Number Contribution
Assignments 0 0
Presentation 0 0
Midterm Examinations (including preparation) 0 0
Project 0 0
Laboratory 0 0
Other Applications 0 0
Quiz 6 10
Term Paper/ Project 0 0
Portfolio Study 0 0
Reports 0 0
Learning Diary 0 0
Thesis/ Project 0 0
Seminar 0 0
Other 0 0
Make-up 0 0
Toplam 6 10
Relation of Proficiency
No Program Learning Outcomes Contribution
1 2 3 4 5
1 understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways; X
2 can properly state and use concepts and results of major mathematical interest; X
3 masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools; X
4 is able to express one’s mathematical ideas in an organised way both in written and oral forms; X
5 understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...); X
6 has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems; X
7 has a theoretical and practical knowledge in computer science well adapted for learning a programming language; X
8 has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions; X
9 has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement; X
10 masters French language as well as other foreign languages, to a level sufficient to study or work abroad. X
ECTS
Activities Number Period Total Workload
Class Hours 70 1 70
Working Hours out of Class 14 4 56
Assignments 0 0 0
Presentation 0 0 0
Midterm Examinations (including preparation) 0 0 0
Project 0 0 0
Laboratory 0 0 0
Other Applications 0 0 0
Final Examinations (including preparation) 0 0 0
Quiz 6 8 48
Term Paper/ Project 0 0 0
Portfolio Study 0 0 0
Reports 0 0 0
Learning Diary 0 0 0
Thesis/ Project 0 0 0
Seminar 0 0 0
Other 0 0 0
Make-up 1 25 25
Total Workload 199
Total Workload / 25 7.96
Credits ECTS 8
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