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.
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
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course accompanied by discussion sessions
|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|
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
final exam 70%, discussion session 30%.
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