Exercise 5.1. Let \(A\) be a subspace of a topological space \(X.\) Prove: if \(F\subseteq A\) and \(F\) is closed in \(X,\) then \(F\) is closed in \(A.\)

Exercise 5.2 (unions and intersections of compact sets). Let \(X\) be a topological space.

• 1. Show that a union of two compact subsets of \(X\) is compact.

• 2. Assuming that \(X\) is Hausdorff, show that an intersection of two compact subsets of \(X\) is compact. (Why do we need \(X\) to be Hausdorff?)

Exercise 5.3 (nested sequence of closed subsets in a compact). Let \(K\) be a compact topological space. Assume that \(F_1\supseteq F_2\supseteq F_3\supseteq \dots ,\) where \(F_i\) is a non-empty closed subset of \(K\) for each \(i\ge 1.\) Prove that all the sets \(F_i\) have a common point.

Exercise 5.4 (a Hausdorff compact topology is “optimal”). Let \((X,\mathscr T)\) be a Hausdorff compact topological space. Use the Topological Inverse Function Theorem to show that

• 1. any topology on \(X,\) which is strictly weaker than \(\mathscr T,\) is not Hausdorff;

• 2. any topology on \(X,\) which is strictly stronger than \(\mathscr T,\) is not compact.