Set and Topology I

Academic unit or major

Mathematics

Instructor(s)

Kotaro Yamada

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-4 Tue

Class

-

Course Code

ZUA.B201

Number of credits

200

Course offered

2024

Offered quarter

1-2Q

Syllabus updated

Mar 17, 2025

Language

Japanese

Syllabus

Course overview and goals

The main subject of this course is basic concepts in set theory and ordered set, Euclidean space and general metric space. 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. Then we introduce the equivalence of sets, and learn the notion of cardinality. 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. We strongly recommend to take this course with "Exercises in Set and Topology".
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 concrete maps
・Be familiar with many basic examples of equivalence relations and quotient sets
・Understand the difference between countable and uncountable sets
・Distinguish between semi-ordered sets and totally-ordered sets
・Be able to deduce strong 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

set, map, image and inverse image, product set, binary relation, equivalence relation, quotient set, cardinality of sets, countable and uncountable set, 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

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)

N/A

Reference books, course materials, etc.

Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.

Evaluation methods and criteria

By term and midterm exams. Details will be explained on the first lecutre.

Related courses

• MTH.B201 ： Introduction to Topology I
• MTH.B202 ： Introduction to Topology II
• ZUA.B202 ： Exercises in Set and Topology

Prerequisites

Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation.
Strongly recommended to take ZUA.B202 ： Exercises in Set and Topology (if not passed yet) at the same time

Contact information (e-mail and phone) Notice : Please replace from ”[at]” to ”@”(half-width character).

kotaro[at]math.titech.ac.jp

Office hours

N/A