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| A set ''E'' is ''compact'' if and only if, for every family [[latex($\{G_{ \alpha } \}_{\alpha \in A}$)]] of open sets such that [[latex2($E \subset \cup_{\alpha \in A}G_{\alpha}$)]] | A set ''E'' is ''compact'' if and only if, for every family [[latex2($\{G_{ \alpha } \}_{\alpha \in A}$)]] of open sets such that [[latex2($E \subset \cup_{\alpha \in A}G_{\alpha}$)]], there is a finite set [[latex2($\{\alpha_1 ,..., \alpha_n \} \subset A$)]] such that [[latex2($E \subset \cup_{i=1}^{n} G_{\alpha_i}$)]]. |
A set E is compact if and only if, for every family latex2($\{G_{ \alpha } \}_{\alpha \in A}$) of open sets such that latex2($E \subset \cup_{\alpha \in A}G_{\alpha}$), there is a finite set latex2($\{\alpha_1 ,..., \alpha_n \} \subset A$) such that latex2($E \subset \cup_{i=1}^{n} G_{\alpha_i}$).
Heine-Borel Theorom: A set latex2(\usepackage{amsfonts} % $E \subset \mathbb{R}$) is compact iff E is closed and bounded.
Example [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.
Introduction to Analysis 5th edition by Edward D. Gaughan