### 2022　Introduction to Topology III

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Instructor(s)
Endo Hisaaki  Kawai Shingo
Class Format
Lecture / Exercise    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(H103)  Tue5-8(W934)
Group
-
Course number
MTH.B203
Credits
2
2022
Offered quarter
3Q
Syllabus updated
2022/9/1
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

The main subject of this course is basic concepts in general topology. First, general topology on sets will be introduced in terms of system open sets, closed sets, and system of neighborhoods, and then continuity for mapping between topological spaces will be discussed. Next we explain various natural topology, such as metric topology, relative topology, quotient topology and product topology. Finally, we discuss various axioms of separability, such as Hausdorff property. This course will be succeeded by “Introduction to Topology IV” in the fourth quarter.
The notion of topological space is essential for describing continuity of mappings. It is significant not only in geometry but also algebra and analysis.

### Student learning outcomes

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

### Keywords

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

### Competencies that will be developed

 ✔ 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/Required learning

Course schedule Required learning
Class 1 topology and topological space Details will be provided during each class session
Class 2 discussion session Details will be provided during each class session
Class 3 open basis, system of neighborhoods, second countability Details will be provided during each class session
Class 4 discussion session Details will be provided during each class session
Class 5 fundamental system of neighborhoods, first countability Details will be provided during each class session
Class 6 discussion session Details will be provided during each class session
Class 7 continuous map, homeomorphism Details will be provided during each class session
Class 8 discussion session Details will be provided during each class session
Class 9 relative topology, product topology Details will be provided during each class session
Class 10 discussion session Details will be provided during each class session
Class 11 quotient topology, induced topology Details will be provided during each class session
Class 12 discussion session Details will be provided during each class session
Class 13 Hausdorff space, normal space Details will be provided during each class session
Class 14 discussion session Details will be provided during each class session

### Out-of-Class Study Time (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.

None required

### Reference books, course materials, etc.

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

### Assessment criteria and methods

final exam 70%, discussion session 30%.

### Related courses

• MTH.B201 ： Introduction to Topology I
• MTH.B202 ： Introduction to Topology II
• MTH.B204 ： Introduction to Topology IV

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Introduction to Topology I and II.
Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation

### Other

T2SCHOLA will be used.