Real Analysis/Compact Sets

Definition

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

Definition

Let (X, d) be a metric space and let A ⊆  X. We say that A is compact if for every open cover {U_λ}_λ∈Λ there is a finite collection U_λ1, …,U_λk 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.

Theorem

Let A be a compact set of a metric space (X, d), then A is closed and bounded.

Theorem (Heine-Borel)

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