Introduction to Topology IV

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 / 5-8 Tue

Class

-

Course Code

MTH.B204

Number of credits

110

Course offered

2024

Offered quarter

4Q

Syllabus updated

Mar 17, 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

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.