1. Manetti 4.22

  2. Manetti 4.25

  3. Manetti 4.26

  4. Manetti 4.27

  5. Let \((X, d)\) be a metric space and \(A \subseteq X\) a set. For any point \(x \in X\), define \(d(x, A)\), the distance from x to \(A\), to be the infimum of the set \(\{d(x, a) \mid a \in A\}\). Show that if \(A\) is compact then for any \(x \in X\) there exists some \(a \in A\) such that \(d(x, A) = d(x, a)\).