COURSE INFORMATION

Course Title Code Semester L+P

Hour Credits ECTS

TOPOLOGY MATH 531 1-2 3 + 0 3 10

Prerequisites -

Language of

Instruction English

Course Level Graduate

Course Type

Course Coordinator Assoc. Prof. Dr. Ender Abadoğlu

Instructors

Assistants

Goals

To provide basic knowledge about topological spaces and their topological properties, to investigate the basic algebraic structures related to the topological spaces.

Content

Definition of topology, topological spaces, continuity, product and subspace topology, connectedness and compactness, countability and separation axioms, The Tychonoff theorem, metrization theorems. Homotopy of paths, homotopy of maps, fundamental group, covering spaces, homotopy lifting property and loop spaces.

Learning Outcomes Teaching

Methods

Assessment

Methods

1) Has ability to analyze topological properties of a space 1 A,B

2) Has ability to relate topological properties and algebraic

structures related to a topological space. 1 A,B

Teaching

Methods: 1: Lecture, 2:Problem solving

Assessment

Methods: A: Written Examination, B: Homework

COURSE CONTENT

Week Topics Study

Materials

1 Topological spaces, basis for a topology, Munkres,

Ch.2.1-3.

2 Product topology on XxY, subspace topology Munkres,

Ch.2.4-5.

3 Continuous functions, product topology, quotient topology Munkres,

Ch.2.6-8, 2.11.

4 Metric topology Munkres,

Ch.2.9-10

5 Connected spaces, path connectedness Munkres,

Ch.3.1-3.

6 Compact spaces Munkres,

Ch.3.5-7.

7 Countability axioms, separation axioms, Munkres,

Ch.4.1-2.

8 Urysohn Lemma, Urysohn metrization theorem, partition of unity Munkres,

Ch.4.3-5.

9 Tychonoff theorem, completely regular spaces, Stone-Cech

compactification

Munkres,

Ch.5.1-3.

10 Homotopy of paths and maps

Greenberg-

Harper, Part I.1-

3.

11 Fundamental group and fundamental group of the circle Greenberg-

Harper, Part I.4.

12 Covering spaces Greenberg-

Harper, Part I.5.

13 Homotopy Lifting Greenberg-Harper, Part I.6.

14 Loop spaces and higher homotopy groups. Greenberg-

Harper, Part I.7

RECOMMENDED SOURCES

Textbook J.R. Munkres,Topology, Second Edition, Prentice-Hall, 2000,

Additional Resources M.J. Greenberg, J.R. Harper,Algebraic Topology: A first course, The

Benjamin/Cummings Publishing Company, 1981.

MATERIAL SHARING

Documents

Assignments

Exams

ASSESSMENT

IN-TERM STUDIES NUMBER PERCENTAGE

Mid-terms

Quizzes

Assignments 5 100

Total 100

CONTRIBUTION OF FINAL EXAMINATION TO OVERALL

GRADE 50

CONTRIBUTION OF IN-TERM STUDIES TO OVERALL

GRADE 50

Total 100

COURSE CATEGORY

COURSE'S CONTRIBUTION TO PROGRAM

No Program Learning Outcomes Contribution

1 2 3 4 5

1 Acquires a rigorous background about the fundamental fields (algebra-analysis-geometry) in mathematics.

x

2 Acquires the ability to relate, interpret, analyse and synthesize on fundamental fields in mathematics and/or mathematics and other sciences.

x

3 Follows contemporary scientific developments, analyses, synthesizes and evaluates novel ideas.

x

4 Uses the national and international academic sources, and computer and related IT.

x

5

Participates in workgroups and research groups, scientific meetings, contacts by oral and written communication at national and international levels.

x

6

Acquires the potential of creative and critical thinking, problem solving,

research, to produce a novel and original work, self-development in areas of interest.

x

7

Acquires the consciousness of scientific ethics and responsibility. Takes

responsibility about the solution of professional problems as a requirement of the intellectual consciousness.

x

ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION

Activities Quantity Duration

(Hour)

Total

Workload

(Hour)

Course Duration (14x Total course hours) 14 3 42

Hours for off-the-classroom study (Pre-study, practice) 14 8 112

Mid-terms (Including self study)

Quizzes

Assignments 5 12 60

Final examination (Including self study) 1 36 36

Total Work Load

250

Total Work Load / 25 (h) 10

ECTS Credit of the Course 10