Real Analysis/Compact Sets

Compact sets are sets ${\displaystyle V}$ for which every open cover, or sequence of open sets that contain ${\displaystyle V}$, contains a finite cover of ${\displaystyle V}$.

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 ${\displaystyle \textstyle A\subseteq \bigcup _{i=1}^{k}U_{\lambda _{i}}}$. 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 has a convergent subsequence.

Every infinite subset K of a compact set V has a limit point in V, as otherwise there would exist an infinite sequence of neighborhoods around points of K that does not contain a finite cover of V.

Every closed subset of a compact set is compact.

Proof: If ${\displaystyle V}$ is closed, and ${\displaystyle V\subset X}$, then ${\displaystyle V\backslash X}$ is open, so the union of any open cover ${\displaystyle A_{n}}$ of ${\displaystyle V}$ and ${\displaystyle V\backslash X}$ is an open cover of ${\displaystyle X}$. Therefore, ${\displaystyle (V\backslash X)\cap A_{n}}$ must contain a finite subcover of ${\displaystyle X}$, and ${\displaystyle A_{n}}$ contains a finite subcover of ${\displaystyle V}$.

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