2016 Introduction to Topology IV

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Academic unit or major
Undergraduate major in Mathematics
Honda Nobuhiro  Nitta Yasufumi  Yamakawa Daisuke 
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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


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
Class 3 connectedness of a topological space
Class 4 discussion session
Class 5 path-connectedness of a topological space
Class 6 discussion session
Class 7 compactness of a topological space
Class 8 discussion session
Class 9 properties of a compact space
Class 10 discussion session
Class 11 completeness of metric spaces
Class 12 discussion session
Class 13 topological properties of metric spaces
Class 14 discussion session
Class 15 evaluation of progress


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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