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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.

X = \bigcup_α U_α ⇒ X = \bigcup_i=1ⁿ U_αᵢ

Definition of compactness

예제 1

Explain why [0, 1] is compact using Heine-Borel.

예제 2

Show (0, 1) is not compact.

Existence of max/min of continuous functions

Compact operators

#컴팩트#열린덮개#compact#open cover