### 2018　Introduction to Topology II

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Instructor(s)
Endo Hisaaki  Kawai Shingo
Class Format
Lecture / Exercise
Media-enhanced courses
Day/Period(Room No.)
Tue3-8(H103)
Group
-
Course number
MTH.B202
Credits
2
2018
Offered quarter
2Q
Syllabus updated
2018/3/20
Lecture notes updated
-
Language used
Japanese
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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

None

### Reference books, course materials, etc.

Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.