Exercise 2.1.10 (Complement of a closure of an open set)

Question

If $U$ is open, then prove that $\overline{\overline{\overline{U}^{\prime}}^{\prime}} = \overline{U}$, where the bar means 	extit{closure} and the prime means 	extit{complement}.

Elu Thingol · Accepted Answer

\begin{proof}
Decoding the notation in the exercise, the theorem assertion becomes
$$\overline{ X \setminus \overline{ X \setminus \overline{U} } } = \overline{U}.$$

Let's look closer at the left-hand side. By \href{./exercise_2-1-9}{Exercise 2.1.9 (a)} we can express it as
\begin{align*}
\overline{ X \setminus \overline{ X \setminus \overline{U} } } &=
X \setminus \operatorname{int}\left(\overline{X \setminus \overline{U}}ight)\
&= X \setminus \operatorname{int}\left(X \setminus \overline{U}ight)
\intertext{Applying \href{./exercise_2-1-9}{Exercise 2.1.9 (b)} we get}
&= X \setminus \left(X \setminus \overline{U}ight)\
&= \overline{U}.
\end{align*}
\end{proof}

Exercise 2.1.10 (Complement of a closure of an open set)

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Exercise 2.1.10 (Complement of a closure of an open set)

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