2025 (Current Year) Faculty Courses School of Computing Undergraduate major in Mathematical and Computing Science
Set and Topology I
- Academic unit or major
- Undergraduate major in Mathematical and Computing Science
- Instructor(s)
- Masaaki Umehara / Zin Arai / Sakie Suzuki / Shinya Nishibata / Shunsuke Tsuchioka
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon / 3-4 Thu
- Class
- -
- Course Code
- MCS.T201
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 1Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The objective of this course is to explain the fundamentals of naive set theory and topology. The main theme of this course is naive set theory. In the first half of this course, lectures on sets, maps, axiom of choice, equivalence relations, and the cardinality of sets are given. In the last half of this course, ordered set, Zorn's lemma, and topological properties of Euclidean spaces and metric spaces are given. This course is aimed to connected to the course "Set and Topology II" in the third quarter.
Course description and aims
(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 naive 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, indexed family of sets, axiom of choice, equivalence relation, quotient set, cardinality of set, countable set, uncountable set, order relation, well-ordered set, Zorn's lemma, Euclidean space, metric space.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
The lectures provide the fundamentals of naive 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/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Logic | Understand the contents covered by the lecture. |
| Class 2 | Sets and set operations | Understand the contents covered by the lecture. |
| Class 3 | mappings and their properties | Understand the contents covered by the lecture. |
| Class 4 | Cartesian products and graphs of mappings | Understand the contents covered by the lecture. |
| Class 5 | Equivalence relations and quotient sets | Understand the contents covered by the lecture. |
| Class 6 | Indexed families of sets | Understand the contents covered by the lecture. |
| Class 7 | Cartesian products of an infinite number of sets and axiom of choice | Understand the contents covered by the lecture. |
| Class 8 | Definition of cardinality of sets and comparison between cardinalities | Understand the contents covered by the lecture. |
| Class 9 | Cardinality of the continuum and continuum hypothesis | Understand the contents covered by the lecture. |
| Class 10 | Ordered sets | Understand the contents covered by the lecture. |
| Class 11 | Zorn's lemma and well-ordered sets | Understand the contents covered by the lecture. |
| Class 12 | Applications of Zorn's lemma and well-ordering theorem | Understand the contents covered by the lecture. |
| Class 13 | Open sets and closed sets in Euclidean spaces | Understand the contents covered by the lecture. |
| Class 14 | Metric spaces and continuous mappings | Understand the contents covered by the lecture. |
Study advice (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)
"Naive Set Theory and General Topology (in Japanese)" by Umehara Masaaki and Ichiki Shunsuke published by Shokabo.
Reference books, course materials, etc.
Specified by the lecturer.
Evaluation methods and criteria
By score of final examination. If you register for "MCS.T202: Exercises in Set and Topology I", its score will be counted as a part of contributions also. Details will be announced in the first lecture.
Related courses
- MCS.T202 ： Exercises in Set and Topology I
Prerequisites
The students are strongly encouraged to take "MCS.T202：Exercises in Set and Topology I", simultaneously.