Definition: A collection \(\{G_\alpha\}\) of open sets is said to be an open cover of the set \(A\) if \(A \subset \cup_\alpha G_\alpha\). A set \(A\) is said to be compact if every open cover of \(A\) contains a finite subcover.
In \(\real^d\), we have the much simpler result that a set \(A\) is compact if and only if \(A\) is closed and bounded (in the sense that there exist \(L, U: L \gl \x \gl U\) for all \(\x \in A\)). For example, the set \(\{\x: a_i \le x_i \le b_i\) for all \(i\}\), where \(a_i < b_i\), is compact.
Definition: A function \(g: \real^d \to \real\) has compact support if there exists a compact set \(C\) such that \(g(\x) = 0\) for all \(\x \notin C\).
Consequences of compactness
Compactness is important because many important properties of continuous functions only hold when the domain is a compact set (i.e., the function has compact support):
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\(g\) continuous \(\implies g\) bounded
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\(g\) continuous \(\implies\) there exist \(\a, \b: g(\a) = \inf g(\x)\), \(g(\b) = \sup g(\x)\) (extreme value theorem)
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\(g\) continuous \(\implies g\) uniformly continuous