Exercise 8.2.11

Question

\begin{enumerate}[label=(\alph*)] 
  \item Show that $E$ is closed if and only if $\overline{E}=E$. Show that $E$ is open if and only if $E^{\circ}=E$.
  \item Show that $\overline{E}^{c}=\left(E^{c}\right)^{\circ}$, and similarly that $\left(E^{\circ}\right)^{c}=\overline{E^{c}}$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

The proof is identical to that for Exercise 3.2.14, which is the same question in the context of $\mathbf{R}$ but whose solution does not use any features unique to $\mathbf{R}$.

Exercise 8.2.11

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Exercise 8.2.11

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