2024 Students Enrolled in or before 2015 School of Science Mathematics
Exercises in Set and Topology
- Academic unit or major
- Mathematics
- Instructor(s)
- Kotaro Yamada / Satoshi Nakamura
- Class Format
- Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-8 Tue / 7-10 Tue
- Class
- -
- Course Code
- ZUA.B202
- Number of credits
- 020
- Course offered
- 2024
- Offered quarter
- 1-2Q
- Syllabus updated
- Mar 17, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This course is an exercise session for the lecture course `Set and Topology I' (ZUA.B201). The materials for exercise are chosen from that course.
Course description and aims
Students are expected to
・Understand De Morgan’s law
・Be familiar with injectivity, surjectivity, and bijectivity of mappings
・Be able to determine the image and preimage of concrete maps
・Be familiar with many basic examples of equivalence relations and quotient sets
・Understand the difference between countable and uncountable sets
・Distinguish between semi-ordered sets and totally-ordered sets
・Be able to deduce strong 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
set, map, image and inverse image, product set, binary relation, equivalence relation, quotient set, cardinality of sets, countable and uncountable set, 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
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Students are given exercise problems related to what is taught in the course "Set and Topology I"
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | discussion session on the following materials: examples of sets, union, intersection and subset, complement | Details will be provided during each class session |
| Class 2 | discussion session on the following materials: De Morgan's law, distributive law, mapping between sets | Details will be provided during each class session |
| Class 3 | discussion session on the following materials: image and preimage of map, composition of maps, product set | Details will be provided during each class session |
| Class 4 | discussion session on the following materials: correspondence between sets, indexed set | Details will be provided during each class session |
| Class 5 | discussion session on the following materials: binary relation, equivalence relation, equivalence class, quotient set | Details will be provided during each class session |
| Class 6 | discussion session on the following materials: the cardinality of set, relation between cardinality, countable set | Details will be provided during each class session |
| Class 7 | discussion session on the following materials: cardinality of the continuum, uncountable set, cardinality of power set | Details will be provided during each class session |
| Class 8 | discussion session on the following materials: order, total order, well-ordered set and their basic properties | Details will be provided during each class session |
| Class 9 | discussion session on the following materials: inductive set, Zorn's lemma | Details will be provided during each class session |
| Class 10 | discussion session on the following materials: ordinal number, comparison of cardinality | Details will be provided during each class session |
| Class 11 | discussion session on the following materials: Equivalence between the well-ordering theorem, Zorn’s lemma and the axiom of choice | Details will be provided during each class session |
| Class 12 | discussion session on the following materials: Application of Zorn's lemma | Details will be provided during each class session |
| Class 13 | discussion session on the following materials: Euclidean space, metric space, open set and closed set | Details will be provided during each class session |
| Class 14 | discussion session on the following materials: basic concepts on metric spaces | Details will be provided during each class session |
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)
N/A
Reference books, course materials, etc.
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
Evaluation methods and criteria
Details will be explained on the first class.
Related courses
- ZUA.B201 ： Set and Topology I
- MTH.B201 ： Introduction to Topology I
- MTH.B202 ： Introduction to Topology II
Prerequisites
Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation.
Strongly recommended to take ZUA.B201 ： Set and Topology I (if not passed yet) at the same time