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Compactness
Undergraduate
Definition
A space is compact if every open cover has a finite subcover. In ℝⁿ, this is equivalent to being closed and bounded.
Formulas
X = \bigcup_α U_α ⇒ X = \bigcup_i=1ⁿ U_αᵢ
Definition of compactness
Examples
Example 1
Explain why [0, 1] is compact using Heine-Borel.
Example 2
Show (0, 1) is not compact.
Applications
Analysis
Existence of max/min of continuous functions
Functional Analysis
Compact operators
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