Exercise 3.3.4

Question

Assume $K$ is compact and $F$ is closed. Decide if the following sets are definitely compact, definitely closed, both, or neither.
  \begin{enumerate}[label=(\alph*)] 
  \item $K \cap F$
  \item $\overline{F^{c} \cup K^{c}}$
  \item $K \backslash F=\{x \in K: x 
otin F\}$
  \item $\overline{K \cap F^c}$
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Compact since $K \cap F$ is closed (finite intersection of closed sets) and bounded (since $K$ is bounded)
  \item Closed but not Compact since $K$ being bounded implies $K^c$ is unbounded, meaning $\overline{F^c \cup K^c}$ is unbounded.
  \item $K \setminus F = K \cap F^c$ could be either, if $K = [0,1], F^c = (0,1)$ then $K \cap F^c$ is open, but if $K = [0,1]$ and $F^c = (-1, 2)$ then $K \cap F^c = [0, 1]$ is compact.
  \item Compact since $K \cap F^c$ is bounded (since $K$ is bounded) implies $\overline{K \cap F^c}$ is closed (closure of a set is closed) and bounded (if $A$ is bounded then $\overline{enumerate}

Exercise 3.3.4

Answers

Comments

Exercise 3.3.4

Answers

Comments

Add answer