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2021　Set and Topology I

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Academic unit or major
Undergraduate major in Mathematical and Computing Science
Instructor(s)
Suzuki Sakie  Umehara Masaaki  Miura Hideyuki  Murofushi Toshiaki  Nishibata Shinya  Tsuchioka Shunsuke 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Mon3-4(W631)  Thr3-4(W631)  
Group
-
Course number
MCS.T201
Credits
2
Academic year
2021
Offered quarter
1Q
Syllabus updated
2021/3/19
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

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

Class flow

The lectures will provide on zoom the fundamentals of set theory and an introduction of topology of metric spaces. Of the two lectures per week, one is live and once is recorded on-demand. Live lessons will also be recorded and URLs will be distributed. We will give feedback and quizzes by Google form every time, and follow up in the next class.
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.

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

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

Reference books, course materials, etc.

The lecture notes will be available online.

Assessment criteria and methods

Evaluation will be made by quizzes and reports. In the quiz, the basic understanding of the lecture content will be asked, and the report will consist of the questions to be considered.

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.

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