2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Introduction to Topology I
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Hisaaki Endo / Satoshi Nakamura
- Class Format
- Lecture/Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue / 7-10 Tue
- Class
- -
- Course Code
- MTH.B201
- Number of credits
- 110
- Course offered
- 2025
- Offered quarter
- 1Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main subject of this course is basic concepts in set theory. After introducing some fundamental operations for sets such as intersection, union and complement, we explain basic notions for maps between sets, such as injection, surjection, and bijection. Next we introduce binary relations on sets, especially the concept of equivalence relation and the associated quotient set. Finally, we introduce the equivalence of sets, and learn the notion of cardinality. Each lecture will be accompanied by a problem solving class. This course will be succeeded by “Introduction to Topology II” in the second 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.
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 maps
・Be familiar with many basic examples of equivalence relations and quotient sets
・Understand the difference between countable and uncountable sets
Keywords
set, map, image and inverse image, product set, binary relation, equivalence relation, quotient set, cardinality of sets, countable and uncountable set
Competencies
- 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/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | examples of sets, union, intersection and subset, complement | Details will be provided during each class session |
| Class 2 | discussion session | Details will be provided during each class session |
| Class 3 | De Morgan's law, distributive law, mapping between sets | Details will be provided during each class session |
| Class 4 | discussion session | Details will be provided during each class session |
| Class 5 | the image and preimage of map, composition of maps, product set | Details will be provided during each class session |
| Class 6 | discussion session | Details will be provided during each class session |
| Class 7 | correspondence between sets, indexed set | Details will be provided during each class session |
| Class 8 | discussion session | Details will be provided during each class session |
| Class 9 | binary relation, equivalence relation, equivalence class, quotient set | Details will be provided during each class session |
| Class 10 | discussion session | Details will be provided during each class session |
| Class 11 | the cardinality of set, relation between cardinality, countable set | Details will be provided during each class session |
| Class 12 | discussion session | Details will be provided during each class session |
| Class 13 | cardinality of the continuum, uncountable set, cardinality of power set | Details will be provided during each class session |
| Class 14 | discussion session | 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)
None
Reference books, course materials, etc.
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
Evaluation methods and criteria
Examination 50%; Exercises 50%
Related courses
- MTH.B202 ： Introduction to Topology II
- MTH.B203 ： Introduction to Topology III
- MTH.B204 ： Introduction to Topology IV
Prerequisites
Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation
Other
Although any textbook is specified, students are encouraged to have a book on sets and topology.