Course Information
Course Title Code Semester T + P ECTS
Metric Spaces II MTK204 4 3 + 0 4

Prerequisites None

Language Turkish
Level Bachelor's Degree
Type Elective
Coordinator Assist.Prof. YÜKSEL SOYKAN
Instructors Assist.Prof. YÜKSEL SOYKAN, Assist.Prof. CAN MURAT DİKMEN, Prof. A. ERDAL COŞKUN, Assoc.Prof. TÜLİN COŞKUN, Assist.Prof. MELİH GÖCEN
Goals The purpose of this course is to give the basic concepts of metric spaces, such as connectedness and compactness.
Contents Connectedness, compactness, finite and infinite metric products, Weierstrass approximation theorem.
Work Placement(s) Absent

Number Learning Outcomes
1 Describes the definitions of connectedness, local connectedness and arcwise connectedness and gives examples of them.
2 Explains and gives examples of the types of compactness.
3 Knows Dini theorem and Arzela-Ascoli theorem and their proofs.
4 Describe finite and infinite metric products and gives examples.
5 Knows Weierstrass approximation theorem and proof.
6 Knows Stone-Weierstrass theorem and its proof.

Mode of Delivery Face-to-Face
Planned Learning Activities & Teaching Methods Lecture, question and answer, discussion, problem solving.
Assessment Methods Midterm exam, homework, final exam.



Course Content
Week Topics Study Materials
1 Connectedness. Studying on related topics from the course materials
2 Local and arcwise connectedness. Studying on related topics from the course materials
3 Topological and ball cover compaktness,totally boundedness. Studying on related topics from the course materials
4 Sequentially compactness, countably compactness. Studying on related topics from the course materials
5 Continuous functions on compact metric spaces. Studying on related topics from the course materials
6 Dini theorem, locally compact spaces, Arzela-Ascoli theorem. Studying on related topics from the course materials
7 Midterm exam. Studying on topics covered in the previous weeks and solving various problems
8 Compact sets in special metric spaces. Studying on related topics from the course materials
9 Finite and Infinite Products of Sets. Finite Metric Products. Studying on related topics from the course materials
10 Infinite Metric Products. Studying on related topics from the course materials
11 Some results in normed spaces. Studying on related topics from the course materials
12 Weierstrass approximation theorem. Studying on related topics from the course materials
13 Stone-Weierstrass theorem. Studying on related topics from the course materials
14 Stone-Weierstrass theorem. Studying on related topics from the course materials
15 Final exam. Studying on topics covered in the previous weeks and solving various problems



Sources
Textbook 1. Yüksel Soykan, "Metrik Uzaylar ve Topolojisi", Nobel Akademik Yayıncılık, 2012. 2. Yüksel Soykan, "Metrik Uzaylar ve Topolojisi Çözümlü Alıştırmaları", Nobel Akademik Yayıncılık, 2012.
Additional Resources 1. Ansari, Q.H., “Metric Spaces", Alpha Science International Ltd, 2010. 2. Giles, J. R., “Introduction to the Analysis of Metric Spaces", Australian Mathematical Society Lecture Series, Cambridge University Press, 1989. 3. Manabendra, N.M., “Metric Spaces", Academic Publishers, Third Edition, Reprint, 2011. 4. Pitts, C.G.C, “Introduction to Metric Spaces", Oliver and Boyd, 1972. 5.Ram, B., “Metric Spaces", Vinayak Publications, 2005. 6.Shirali, S., Vasudeva, H.L., “Metric Spaces”, Springer, 2006. 7.Tyagi, B.K., “Metric Spaces", Cambridge University Press, India Pvt. Ltd. 2010.



Assessment System Quantity Percentage
In-Term Studies
Mid-terms 1 70
Assignments 1 30
In-Term Total 2 100
Contribution of In-Term Studies to Overall 40
Contribution of Final Exam to Overall 60
Total 100





Course's Contribution to PLO
No Key Learning Outcomes Level
1 2 3 4 5
1 Has textbooks containing current information, application tools and equipment , and advanced theoretical and practical knowledge supported by other resources in a scientific approach. x
2 Adapts and transfers the acquired knowledge to secondary education. x
3 Uses advanced institutional and practical knowledge acquired in the field. x
4 Updates the information on daily conditions. x
5 Comments on and evaluate the data by using advanced knowledge and skills acquired in the field, identifies and analyzes the current problems of technological developments, and comes up with solutions based on research and evidence. x
6 Has the ability to conceptualize the events and facts related with the field; analyze them with scientific methods and techniques. x
7 Designs and performs experiments to analyze the problems, collects data, performs analyzes and comment on the results.
8 Plans and manages the activities in a project under his responsibility for development.
9 Plays a role in the process of decision making when faced with problems about different discipline fields.
10 Uses time effectively in the process of inference with the ability of thinking analytically.
11 Evaluates the advanced knowledge and skills acquired in the field with a critical perspective.
12 Determines the learning requirements and leads the learning process.
13 Develops a positive attitude towards lifelong learning.
14 Is aware of the necessity of lifelong learning and develops his Professional knowledge and skills continuously.
15 Informs people and organizations about the topics related to their fields; expresses his ideas and suggestions for solutions to problems in both oral and written form.
16 Shares his ideas and suggestions for solutions to the problems with experts and non-experts by supporting them with quantitative and qualitative data.
17 Organizes projects and activities for social environment he lives in with an awareness of social responsibility.
18 Follows advances in the field and communicate with colleagues by using a foreign language at least at B1 level of European Language Portfolio.
19 Uses information and communication technology along with software the Human Sciences the field requires at an advanced level.
20 Uses his knowledge of human health and environmental awareness acquired in their fields for society’s ends.
21 Behaves in a way adhering to the social, scientific, cultural and ethical values in the process of data collection, commenting, application, publicizing the results related with the field.
22 Has a sufficient level of awareness about the universality of social rights, social justice, quality management, acting in a suitable way in processes and attendance (Instead of quality culture) the protection of cultural values, protection of the environment and health and security in the professional field.
23 (Waiting for translation) Alanı ile ilgili verilerin toplanması, yorumlanması, uygulanması ve sonuçlarının duyurulması aşamalarında toplumsal, bilimsel, kültürel ve etik değerlere uygun hareket eder.
24 (Waiting for translation) Sosyal hakların evrenselliği, sosyal adalet, kalite yönetimi ve süreçlerine uygun davranma ve katılma (Kalite kültürünün yerine) ve kültürel değerlerin korunması ile çevre koruma, iş sağlığı ve güvenliği konularında yeterli bilince sahiptir.



ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION
Activities Quantity Duration (Hour) Total Work Load (h)
Course Duration 14 3 42
Hours for off-the-classroom study (Pre-study, practice) 14 2 28
Assignments 1 10 10
Mid-terms 1 20 20
Final examination 1 30 30
Total Work Load (h) 130
Total Work Load / 30 (h) 4.33
ECTS Credit of the Course 4