2020 Introduction to Topology IV

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Gomi Kiyonori  Masai Hidetoshi 
Course component(s)
Lecture / Exercise    (ZOOM)
Day/Period(Room No.)
Tue3-8(H103)  
Group
-
Course number
MTH.B204
Credits
2
Academic year
2020
Offered quarter
4Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

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.

Student learning outcomes

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 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 separation axioms and continuous functions Details will be provided during each class session
Class 2 discussion session Details will be provided during each class session
Class 3 connectedness of a topological space Details will be provided during each class session
Class 4 discussion session Details will be provided during each class session
Class 5 path-connectedness of a topological space Details will be provided during each class session
Class 6 discussion session Details will be provided during each class session
Class 7 compactness of a topological space Details will be provided during each class session
Class 8 discussion session Details will be provided during each class session
Class 9 properties of a compact space Details will be provided during each class session
Class 10 discussion session Details will be provided during each class session
Class 11 completeness of metric spaces Details will be provided during each class session
Class 12 discussion session Details will be provided during each class session
Class 13 topological properties of metric spaces Details will be provided during each class session
Class 14 discussion session Details will be provided during each class session
Class 15 evaluation of progress 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.

Textbook(s)

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.B203 : Introduction to Topology III
  • MTH.B211 : Introduction to Geometry I
  • MTH.B212 : Introduction to Geometry II

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

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

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