Introduction to Topology II

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)

3-8 Tue

Class

-

Course Code

MTH.B202

Number of credits

110

Course offered

2025

Offered quarter

2Q

Syllabus updated

Mar 19, 2025

Language

Japanese

Syllabus

Course overview and goals

The main subjects of this course are the basic concepts of ordered set, Euclidean space and general metric space. After introducing the basic notions of ordered set, well-ordered set, and inductive set, some applications of these concepts will be provided. We also introduce Euclidean space and learn that the continuity of maps between Euclidean spaces can be simply rephrased by making use of open sets. Finally, we discuss the notion of general metric space, and learn that the continuity of maps between them may also be simply described using open sets. This course is a succession of “Introduction to Topology I” in the first 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
・Distinguish between semi-ordered sets and totally-ordered sets
・Be able to deduce particular 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

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

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

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.B201 ： Introduction to Topology I
• MTH.B203 ： Introduction to Topology III
• MTH.B204 ： Introduction to Topology IV

Prerequisites

Students are required to have passed Introduction to Topology I.
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.