Real Analysis/Compact Sets

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Definition[edit]

Let (X, d) be a metric space and let A ⊆  X. A collection of open sets {Uλ}λ∈Λ is called an open cover of A if A ⊆ λ∈Λ Uλ.

Definition[edit]

Let (X, d) be a metric space and let A ⊆  X. We say that A is compact if for every open cover {Uλ}λ∈Λ there is a finite collection Uλ1, …,Uλk so that \textstyle A\subseteq \bigcup_{i=1}^k U_{\lambda_i}. In other words a set is compact if and only if every open cover has a finite subcover. There is also a sequential definition of compact set. A set A in the metric space X is called compact if every sequence in that set have a convergent subsequence.

Theorem[edit]

Let A be a compact set in R^n with usual metric, then A is closed and bounded.

Theorem (Heine-Borel)[edit]

If \textstyle X = \mathbb{R}^n, with the usual metric, then every closed and bounded subset of X is compact.