Decide whether the following propositions are true or false. If the claim is valid, supply a short proof, and if the claim is false, provide a counterexample.
(a) The arbitrary intersection of compact sets is compact.
(b) The arbitrary union of compact sets is compact.
(c) Let A be arbitrary and let K be compact, then the intersection $A\bigcap K$ is compact.
(d) If $F_{1}\supseteq F_{2} \supseteq F_{3} \supseteq F_{4} \cdot\cdot \cdot\cdot $is a nested sequence of nonempty closed sets, then the intersection $\bigcap_{n=1}^\infty F_{n}\neq \emptyset$
For (a-c) I would like to have my attempt for these solutions checked. Please let me know if these are accurate, not accurate, or accurate but not sufficient. For (d) I need more assistance.
My attempt:
(d) Following the nested interval theorem, this holds. Can someone show me how it does or if I am wrong why it doesn't.
Best Answer
For c, you are wrong. Take for example the set $A=(\frac{1}{2},1)$ and $K=[0,2]$. Then their intersection is just $A$ which is not compact. As for d), you need a complete metric space, so in the general case it isn't true.