Set and Topology II

Academic unit or major

Mathematics

Instructor(s)

Hisaaki Endo

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-4 Tue

Class

-

Course Code

ZUA.B203

Number of credits

200

Course offered

2024

Offered quarter

3-4Q

Syllabus updated

Mar 17, 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 of open sets, closed sets, and system of neighborhoods, and then continuity for mapping between topological spaces will be discussed in terms of these notions. Next we explain various natural topology, such as metric topology, relative topology, quotient topology and product topology. Then we discuss various axioms of separability, such as Hausdorff property. Main subjects in the latter half of this course 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. We strongly recommend to take this course with "Exercises in Geometry A".
The notion of topological space is essential for describing continuity of mappings. It is significant not only in geometry but also algebra and analysis. Compactness and connectedness are most significant geometric properties of the space. They will be fundamental when learning more advanced geometry, such as theory of manifolds. Completeness and boundedness of metric spaces are fundamental concepts especially in analysis.

Course description and aims

Students are expected to
・Understand various equivalent definitions of topology
・Understand that continuity of maps between topological spaces is described in terms of topology
・Understand various kinds of topologies that naturally arise under various settings
・Understand various separation axioms, with various examples
・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

topology and topological space, neighborhood, first countability, second countability, continuous mapping, induced topology, separation axioms, 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

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

midterm exam (about 50%), final exam (about 50%)

Related courses

• ZUA.B204 ： Exercises in Geometry A
• MTH.B203 ： Introduction to Topology III
• MTH.B204 ： Introduction to Topology IV

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.B204 ： Exercises in Geometry A (if not passed yet) at the same time