Topology

Explore the fundamental concepts of topology in this comprehensive 10-hour course. Delve into metric spaces, convergence, completeness, open and closed sets, compactness, continuity, and connectedness. Examine intriguing topics like the Cantor set, Baire Category Theorem, and the Topologist Sine Curve. Analyze the Heine Borel Theorem, sequential compactness, and the finite intersection property. Investigate continuity in Rn and topology, and understand homeomorphisms. Conclude with a challenging UC Berkeley Math PhD Entrance Exam question to test your grasp of topological concepts.

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.