2016 Set and Topology II

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Honda Nobuhiro 
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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 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


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
Class 3 fundamental system of neighborhoods, first countability
Class 4 continuous map, homeomorphism
Class 5 relative topology, product topology
Class 6 quotient topology, induced topology
Class 7 Hausdorff space, normal space
Class 8 evaluation of progress
Class 9 separation axioms and continuous functions
Class 10 connectedness of a topological space
Class 11 path-connectedness of a topological space
Class 12 compactness of a topological space
Class 13 properties of a compact space
Class 14 completeness of metric spaces
Class 15 topological properties of metric spaces


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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