Compact Metric Spaces
Sequential Compactness
Compactness in Metric Spaces
Total Boundedness
Heine-Borel Theorem
Properties of Compact Metric Spaces
Extreme Value Theorem
Uniform Continuity
Lebesgue Number
ArzelΓ -Ascoli Theorem
Sequential Compactness and Completeness
Examples
Example 1: is compact in (closed and bounded).
Example 2: is compact ( in ).
Example 3: is not compact (unbounded).
Example 4: is not compact (not closed).
Covering Properties
Compactness and Continuity
Tychonoff's Theorem
(This requires the Axiom of Choice)
Applications
- Existence of solutions to equations
- Optimization on compact domains
- Fixed point theorems
- Functional analysis (compact operators)
- Differential equations (existence of solutions)