Continuous Functions on Compact Sets

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Explores the Stone-Weierstrass theorem, proving uniform density of specific function families on compact sets.

Covers the preliminaries in measure theory, including loc comp, separable, complete metric space, and tightness concepts.

Explores the necessity of uniform continuity for continuous functions on compact sets.

Covers the Cantor-Heine theorem, discussing uniform continuity and compactness.

Explores the Jordan Curve Theorem, illustrating the division of a plane by a simple closed curve into two regions.

Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.

Explores the proof of Weyl's theorem, focusing on discrete spectrum, ground states, and potential energy continuity.

Discusses separation criteria and quotient spaces in topology, emphasizing their applications and theoretical foundations.

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Explores limits, minimum and maximum values of functions, and continuity criteria in a compact set.