Exercise 3.2.8

Question

Assume $A$ is an open set and $B$ is a closed set. Determine if the following sets are definitely open, definitely closed, both, or neither.
  \begin{enumerate}[label=(\alph*)] 
  \item $\overline{A \cup B}$
  \item $A \backslash B=\{x \in A: x 
otin B\}$
  \item $\left(A^{c} \cup Bight)^{c}$
  \item $(A \cap B) \cup\left(A^{c} \cap Bight)$
  \item $\overline{A}^{c} \cap \overline{A^{c}}$
   \end{enumerate}

Ulisse Mini · Accepted Answer

For all of these keep in mind the only open and closed sets are $\mathbf{R}$ and $\emptyset$, and if $A$ is open $A^c$ is closed and vise versa.
  \begin{enumerate}[label=(\alph*)] 
  \item Closed, since the closure of a set is closed.
  \item Open since $B$ being closed implies $B^c$ is open and thus $A \cap B^c$ is open as it is an intersection of open sets.
  \item De Morgan's laws give $(A^c \cup B)^c = A \cap B^c$ which is the same as (b)
  \item Closed since $(A \cap B) \cup (A^c \cap B) = B$
  \item Neither in general. Note that $\overline{A}^c 
e \overline{A^c}$ consider how $A = \{1/n : n \in \mathbf{N}\}$ has $\overline{A^c} = \mathbf{R}$ but $\overline{A}^c 
e \mathbf{R}$.
   \end{enumerate}

Exercise 3.2.8

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Exercise 3.2.8

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