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 $(X,\mathcal{T})$ 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 $\mathcal{T},$ is not Hausdorff;

• 2. any topology on $X,$ which is strictly stronger than $\mathcal{T},$ is not compact.