Topology

This course introduces topology through metric spaces, covering open and closed sets, convergence, completeness, compactness, continuity, connectedness, and homeomorphisms.

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.