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 $$G_n = \left(\frac{1}{n},2\right)$$. If $$0<x \leq 1$$, there is a positive integer n such that $$\frac{1}{n} < x$$; hence, $$x \in G_n$$, and thus
$$$E \subset \bigcup_{n=1}^{\infty}G_n$$$
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
$$$E \not\subset G_{n_0}=\left(\frac{1}{n_0},2\right)$$$
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.