2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Introduction to Topology III
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Kiyonori Gomi / Satoshi Nakamura
- Class Format
- Lecture/Exercise
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.B203
- Number of credits
- 110
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main subject of this course is basic concepts in general topology. First, general topology on sets will be introduced in terms of system open sets, closed sets, and system of neighborhoods, and then continuity for mapping between topological spaces will be discussed. Next we explain various natural topology, such as metric topology, relative topology, quotient topology and product topology. Finally, we discuss various axioms of separability, such as Hausdorff property. This course will be succeeded by “Introduction to Topology IV” in the fourth quarter.
The notion of topological space is essential for describing continuity of mappings. It is significant not only in geometry but also algebra and analysis.
Course description and aims
Students are expected to
・Understand various equivalent definitions of topology
・Understand that continuity of maps between topological spaces are described in terms of topology
・Understand various kinds of topologies that naturally arises under various settings
・Understand various separation axioms, with various examples
Keywords
topology and topological space, neighborhood, first countability, second countability, continuous mapping, induced topology, separation axioms
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 | topology and topological space | Details will be provided during each class session |
| Class 2 | discussion session | Details will be provided during each class session |
| Class 3 | open basis, system of neighborhoods, second countability | Details will be provided during each class session |
| Class 4 | discussion session | Details will be provided during each class session |
| Class 5 | fundamental system of neighborhoods, first countability | Details will be provided during each class session |
| Class 6 | discussion session | Details will be provided during each class session |
| Class 7 | continuous map, homeomorphism | Details will be provided during each class session |
| Class 8 | discussion session | Details will be provided during each class session |
| Class 9 | relative topology, product topology | Details will be provided during each class session |
| Class 10 | discussion session | Details will be provided during each class session |
| Class 11 | quotient topology, induced topology | Details will be provided during each class session |
| Class 12 | discussion session | Details will be provided during each class session |
| Class 13 | Hausdorff space, normal space | 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 required
Reference books, course materials, etc.
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
Evaluation methods and criteria
final exam 60%, discussion session 40%.
Related courses
- MTH.B201 ： Introduction to Topology I
- MTH.B202 ： Introduction to Topology II
- MTH.B204 ： Introduction to Topology IV
Prerequisites
Students are expected to have passed Introduction to Topology I and II.
Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation
Other
T2SCHOLA will be used.