Exercise 3.3.5

Question

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.
  \begin{enumerate}[label=(\alph*)] 
  \item The arbitrary intersection of compact sets is compact.
  \item The arbitrary union of compact sets is compact.
  \item Let $A$ be arbitrary, and let $K$ be compact. Then, the intersection $A \cap K$ is compact.
  \item If $F_{1} \supseteq F_{2} \supseteq F_{3} \supseteq F_{4} \supseteq \cdots$ is a nested sequence of nonempty closed sets, then the intersection $\bigcap_{n=1}^{\infty} F_{n} 
eq \emptyset$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item True, as it will be bounded and closed (since arbitrary intersections of closed sets are closed).
  \item False, $\bigcup_{n=1}^\infty [0, n]$ is unbounded and thus not compact.
  \item False, let $K = [0,1]$ and $A = (0,1)$. The intersection $K \cap A = (0,1)$ is not compact.
  \item False as $\bigcap_{n=1}^\infty [n, \infty) = \emptyset$ (It is true for compact sets though)
   \end{enumerate}

Exercise 3.3.5

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Exercise 3.3.5

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