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- Liberal arts and basic science courses
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Endo Hisaaki Kawai Shingo Nitta Yasufumi
- Class Format
- Lecture / Exercise
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-8(H103)
- Group
- -
- Course number
- MTH.B202
- Credits
- 2
- Academic year
- 2017
- Offered quarter
- 2Q
- Syllabus updated
- 2017/3/17
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
The main subjects of this course are the basic concepts of ordered set, Euclidean space and general metric space. After introducing the basic notions of ordered set, well-ordered set, and inductive set, some applications of these concepts will be provided. We also introduce Euclidean space and learn that the continuity of maps between Euclidean spaces can be simply rephrased by making use of open sets. Finally, we discuss the notion of general metric space, and learn that the continuity of maps between them may also be simply described using open sets. This course is a succession of “Introduction to Topology I” in the first quarter.
The notions of set and map are fundamental not only in mathematics but also in science, and are applicable to describe a wide variety of objects. On the other hand, these abstract notions are not easy to comprehend without suitable training. To that end, rigorous proofs will be provided for most propositions, lemmas and theorems.
Student learning outcomes
Students are expected to
・Distinguish between semi-ordered sets and totally-ordered sets
・Be able to deduce particular properties of well-ordered sets
・Understand a few applications of Zorn’s lemma
・Understand equivalence between the well-ordering theorem, Zorn’s lemma and the axiom of choice
・Understand basic properties of Euclidean space and general metric spaces
Keywords
ordered set, totally ordered set, well-ordered set, Zorn’s lemma, the axiom of choice, well-ordering theorem, Euclidean space, metric space, continuous map
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course accompanied by discussion sessions
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | order, total order, well-ordered set and their basic properties | Details will be provided during each class session |
| Class 2 | discussion session | Details will be provided during each class session |
| Class 3 | inductive set, Zorn's lemma | Details will be provided during each class session |
| Class 4 | discussion session | Details will be provided during each class session |
| Class 5 | Equivalence between the well-ordering theorem, Zorn’s lemma and the axiom of choice | Details will be provided during each class session |
| Class 6 | discussion session | Details will be provided during each class session |
| Class 7 | ordinal number, comparison of cardinality | Details will be provided during each class session |
| Class 8 | discussion session | Details will be provided during each class session |
| Class 9 | Application of Zorn's lemma | Details will be provided during each class session |
| Class 10 | discussion session | Details will be provided during each class session |
| Class 11 | Euclidean space, metric space, open set and closed set | Details will be provided during each class session |
| Class 12 | discussion session | Details will be provided during each class session |
| Class 13 | Basic concepts on metric spaces | Details will be provided during each class session |
| Class 14 | discussion session | Details will be provided during each class session |
| Class 15 | evaluation of progress | Details will be provided during each class session |
Textbook(s)
None
Reference books, course materials, etc.
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
Assessment criteria and methods
final exam (about 70%), discussion session (about 30%).
Related courses
- MTH.B201 ： Introduction to Topology I
- MTH.B203 ： Introduction to Topology III
- MTH.B204 ： Introduction to Topology IV
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are required to have passed Introduction to Topology I.
Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation
