Discussion
Let \(X\) be a topological space and \(A \subseteq X\). Given a subset \(S\) of \(X\), let \(\text{int}(S)\), \(\partial S\), and \(\overline{S}\) denote the interior, boundary, and closure, respectively, of the set \(S\). Which one of the following is NOT necessarily true?
Answer: C
The statement that need not hold is \(\partial A \subseteq \partial(\text{int}(A))\).
In \(\mathbb{R}\), take \(A = \{0\}\). Its interior is empty, so \(\partial(\text{int}(A)) = \emptyset\), but \(\partial A = \{0\}\). So the inclusion fails.
The others always hold. \(\text{int}(X \setminus A)\) is an open set missing \(A\), so it misses \(\overline{A}\) too. \(A \subseteq \overline{A}\) by definition. And since \(\text{int}(A) \subseteq \text{int}(\overline{A})\),
\[ \begin{aligned} \partial\overline{A} &= \overline{A} \setminus \text{int}(\overline{A}) \\ &\subseteq \overline{A} \setminus \text{int}(A) = \partial A. \end{aligned} \]
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