2016 Introduction to Topology III

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Academic unit or major
Undergraduate major in Mathematics
Honda Nobuhiro  Nitta Yasufumi  Kan Toru 
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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


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
Class 3 open basis, system of neighborhoods, second countability
Class 4 discussion session
Class 5 fundamental system of neighborhoods, first countability
Class 6 discussion session
Class 7 continuous map, homeomorphism
Class 8 discussion session
Class 9 relative topology, product topology
Class 10 discussion session
Class 11 quotient topology, induced topology
Class 12 discussion session
Class 13 Hausdorff space, normal space
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.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

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