### 2019　Set and Topology I

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Umehara Masaaki  Nishibata Shinya  Miura Hideyuki  Murofushi Toshiaki  Suzuki Sakie  Tsuchioka Shunsuke
Course component(s)
Lecture
Day/Period(Room No.)
Mon3-4(W833)  Thr3-4(W833)
Group
-
Course number
MCS.T201
Credits
2
2019
Offered quarter
1Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

The objective of this course is to explain the fundamentals of set theory and topology. The main theme of this cousrse is set theory. In the first half of this course, lectures on sets, maps, axiom of chice, equivalence relations, and the cardinarity of sets are given. In the the last half of this course, orderd set, Zorn's lemma, and the topological properties of metric spaces are given. This course is aimed to connected to the course "Set and Topology II" in the third quarter.

### Student learning outcomes

(Theme)
The set and topology play an important role in mathematical and computing science. The objective of this course is to explain the fundamentals of set theory,
and topological properties of metric spaces as an introduction of topology.
(The goal)
The students are expected to understand the fundamentals of mathematical methods to handle the concept of sets, maps, and metric spaces appeared in mathematical and computing science and also to be able to apply them to practical problems.

### Keywords

set, mapping, family of sets indexed by a set, the axiom of choice, equivalence relations, quotient set, cardinality of sets, countabe set, uncountable set, order, linearly ordered set, Zorn's lemma, Euclidean space, metric space.

### Competencies that will be developed

 Intercultural skills Communication skills ✔ Specialist skills Critical thinking skills ✔ Practical and/or problem-solving skills

### Class flow

The lectures provide the fundamentals of set theory and an introduction of topology of metric spaces. The students are strongly encouraged to register for "MCS.T202：Exercises in Set and Topology I" simultaneously which offers the recitation session for this course.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Definition and examples of sets Understand the contents covered by the lecture.
Class 2 Several operations on sets Understand the contents covered by the lecture.
Class 3 mappings and their properties Understand the contents covered by the lecture.
Class 4 family of sets indexed by a set, the axiom of choice Understand the contents covered by the lecture.
Class 5 operations on sets and mappings, equivalence relations and quotient sets Understand the contents covered by the lecture.
Class 6 the definition of cardinality of sets Understand the contents covered by the lecture.
Class 7 Comparison of the cardinarity Understand the contents covered by the lecture.
Class 8 counable sets Understand the contents covered by the lecture.
Class 9 The cardinarity of the continuum and uncountable sets Understand the contents covered by the lecture.
Class 10 orderrings Understand the contents covered by the lecture.
Class 11 linearly ordered set Understand the contents covered by the lecture.
Class 12 Zorn's lemma Understand the contents covered by the lecture.
Class 13 Euclidean space Understand the contents covered by the lecture.
Class 14 topological properties of Euclidean space Understand the contents covered by the lecture.
Class 15 metric spaces Understand the contents covered by the lecture.

### Textbook(s)

"Set and Topological Space (in Japanese)" by Shigeyuki Morita published by Aasakura Shoten.

### Reference books, course materials, etc.

Not specified in particular.

### Assessment criteria and methods

By scores of report (or the midterm examination) and the final examination.

### Related courses

• MCS.T202 ： Exercises in Set and Topology I

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

The students are encouraged to take "MCS.T202：Exercises in Set and Topology I", simultaneously.