2021 Set and Topology II

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Academic unit or major
Mathematics
Instructor(s)
Gomi Kiyonori 
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Tue3-4(H103)  Tue3-4(H112)  
Group
-
Course number
ZUA.B203
Credits
2
Academic year
2021
Offered quarter
3-4Q
Syllabus updated
2021/3/19
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 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.

Student learning outcomes

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 that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
Class 1 topology and topological space Details will be provided during each class session
Class 2 open basis, system of neighborhoods, second countability Details will be provided during each class session
Class 3 fundamental system of neighborhoods, first countability Details will be provided during each class session
Class 4 continuous map, homeomorphism Details will be provided during each class session
Class 5 relative topology, product topology Details will be provided during each class session
Class 6 quotient topology, induced topology Details will be provided during each class session
Class 7 Hausdorff space, normal space Details will be provided during each class session
Class 8 separation axioms and continuous functions Details will be provided during each class session
Class 9 connectedness of a topological space Details will be provided during each class session
Class 10 path-connectedness of a topological space Details will be provided during each class session
Class 11 compactness of a topological space Details will be provided during each class session
Class 12 properties of a compact space Details will be provided during each class session
Class 13 completeness of metric spaces Details will be provided during each class session
Class 14 topological properties of metric spaces 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

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

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

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