MATH535 TOPOLOGY
| Course Code: | 2360535 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. SÜLEYMAN ÖNAL |
| Offered Semester: | Fall Semesters. |
Course Objectives
At the end of this course, the student will learn topological spaces and continuous functions, connected and compact spaces, countability and separation axioms.
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Define and work with topological spaces, open sets, closed sets, bases, and subbases.
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Explore examples of topological spaces (e.g., metric spaces, discrete and indiscrete topologies, finite complements, product topology).
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Define continuous functions in topological terms (via inverse images of open sets).
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Examine homeomorphisms (topological isomorphisms) and topological invariants.
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Learn about Hausdorff spaces, regular spaces, normal spaces, and other separation properties.
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Study first-countable and second-countable spaces.
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Define compact spaces (using open covers) and study key theorems (e.g., Heine-Borel Theorem, Tychonoff’s Theorem for finite products).
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Examine connected and path-connected spaces.Construct new topological spaces using product topologies and quotient topologies.
Course Content
Topological spaces. Neighborhoods. Basis. Subspace topology, product and quotient topologies. Compactness. Tychonoff's Theorem. Heine-Borel theorem. Separation properties. Urysohn's Lemma and Tietze Extension theorem. Stone-Cech compactification. Alexandroff one point compactification. Convergence of sequences and nets. Connectedness. Metrizability. Complete metric spaces. Baire's theorem.
Course Learning Outcomes
Student, who passed the course satisfactorily will be able to solve problems on basic concepts of point set topology.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution