2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Introduction to Topology IV
- 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.B204
- Number of credits
- 110
- Course offered
- 2025
- Offered quarter
- 4Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This course is a succession of “Introduction to Topology III” in 3Q. Main subjects are geometric properties of topological spaces, such as compactness, (path-) connectedness. Compact spaces have distinguished property that any function has maximum and minimum, and one of the fundamental properties of a space. A number of significant examples of compact/ non-compact and connected/disconnected spaces are provided. Also completeness and boundedness of metric spaces are treated.
Compactness and connectedness are most significant geometric properties of the space. They will be fundamental when learning more advanced geometry, such as manifolds. Completeness and boundedness are fundamental concepts especially in analysis.
Course description and aims
Students are expected to
・Be able to prove basic properties of connected and compact spaces
・Learn a lot of basic examples of compact/ non-compact and connected/disconnected spaces
・Understand basic properties of complete metric spaces and examples
Keywords
compact space, connected spaces, path-connectedness, completeness of a metric space
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 | separation axioms and continuous functions | Details will be provided during each class session |
| Class 2 | discussion session | Details will be provided during each class session |
| Class 3 | connectedness of a topological space | Details will be provided during each class session |
| Class 4 | discussion session | Details will be provided during each class session |
| Class 5 | path-connectedness of a topological space | Details will be provided during each class session |
| Class 6 | discussion session | Details will be provided during each class session |
| Class 7 | compactness of a topological space | Details will be provided during each class session |
| Class 8 | discussion session | Details will be provided during each class session |
| Class 9 | properties of a compact space | Details will be provided during each class session |
| Class 10 | discussion session | Details will be provided during each class session |
| Class 11 | completeness of metric spaces | Details will be provided during each class session |
| Class 12 | discussion session | Details will be provided during each class session |
| Class 13 | topological properties of metric spaces | Details will be provided during each class session |
| Class 14 | discussion session | Details will be provided during each class session |
| Class 15 | evaluation of progress | 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.B203 ： Introduction to Topology III
- MTH.B211 ： Introduction to Geometry I
- MTH.B212 ： Introduction to Geometry II
Prerequisites
Required to have passed Introduction to Topology III.
Expected to have passed Introduction to Topology I and II.
Expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation
Other
T2SCHOLA will be used.