Chapter 7.2, Problem 21E

(a)

To determine

To prove: That ${\displaystyle \underset{\alpha \in \Delta }{\cap }{A}_{\alpha }}$ is compact when $\left\{A_{\alpha }:\alpha \in \Delta \right\}$ is non-empty collection of compact sets.

(b)

To determine

To Find: The example of a collection $\left\{A_{\alpha }:\alpha \in \Delta \right\}$ of compact sets such that ${\displaystyle \underset{\alpha \in \Delta }{\cup }{A}_{\alpha }}$ is not compact.