This course covers the fundamentals of Topology, including topics such as metric spaces, convergence, completeness, compactness, continuity, connectedness, and homeomorphism. By the end of the course, students will be able to understand and apply concepts related to open and closed sets, compactness properties, continuity in different spaces, and topological transformations. The course is designed for individuals interested in advanced mathematics and theoretical concepts in topology.
Overview
Syllabus
What is a metric space ?.
Can a disk be a square ?.
Convergence in Rn.
Rn is complete.
Multidimensional Bolzano Weierstraß.
Completion of a metric space.
Taste of topology: Open Sets.
What is a closed set ?.
Can a ball be a sphere?.
Cantor Intersection Theorem.
Cantor set.
Baire Category Theorem.
Compactness.
Compactness.
Properties of Compactness.
Heine Borel Theorem.
[a,b] is compact.
Non Compact set.
Sequential Compactness.
Totally Bounded.
Finite Intersection Property.
Continuity in Rn.
Is addition continuous?.
Continuity in Topology.
f implies continuous.
Continuity and Compactness.
Connectedness.
R is connected.
Topologist Sine Curve.
What is a Homeomorphism.
UC Berkeley Math PhD Entrance Exam Question.
Taught by
Dr Peyam
