CompactSet

Definition A set E is compact if and only if, for every family $\{G_{\alpha }{\}}_{\alpha \in A}$ of open sets such that $E\subset \bigcup _{\alpha \in A}{G}_{\alpha }$, there is a finite set $\{{\alpha }_1,...,{\alpha }_n\}\subset A$ such that $E\subset \bigcup _{i=1}^n{G}_{{\alpha }_i}$.

Example: Let E=(0,1] and for each positive integer n, let $G_n=(\frac{1}{n},2)$. If $0<x\le 1$, there is a positive integer n such that $\frac{1}{n}<x$; hence, $x\in G_n$, and thus

$$E\subset \bigcup _{n=1}^{\mathrm{\infty }}{G}_n$$

If we choose a finite set $n_1,...,n_r$ of positive integers, then

$$\bigcup _{i=1}^r{G}_{n_i}=G_{n_0}$$

where $n_0=max\{n_1,...,n_r\}$ and

$$E\not\subset G_{n_0}=(\frac{1}{n_0},2)$$

Thus, we have a family of open sets $\{G_n{\}}_{n\in J}$ such that $E\subset \bigcup _{n\in J}{G}_n$, but no finite subfamily has this property. From the definition, it is clear that E is not compact.

Heine-Borel Theorom: A set $E\subset \mathbb{R}$ is compact iff $E$ is closed and bounded.

Examples:

• [2,8] is a compact set.
• The unit disk including the boundary is a compact set.
• (3,5] is not a compact set.

Note that all of these examples are of sets that are uncountably infinite.

Definitions from: Introduction to Analysis 5th edition by Edward D. Gaughan

Theorom: The union of compact sets is compact.