Let $X$ be a topological space, and let $S$ be a subset of $X$. Then $\mathrm{int} S$, by definition, is the union of all the open sets of $X$ that are contained in $S$.
Now let $A$ and $B$ be any subsets of $X$. Then what is the relation, if any, between (i) $\mathrm{Int} (A \cup B)$ and $\mathrm{Int} A \cup \mathrm{Int} B$? (ii) $\mathrm{Int} (A \cap B)$ and $\mathrm{Int} A \cap \mathrm{Int} B$? (iii) $\mathrm{Int} (A \setminus B)$ and $\mathrm{Int} A \setminus \mathrm{Int} B$? (iv) $\mathrm{Int} (A \Delta B)$ and $\mathrm{Int} A \Delta \mathrm{Int} B$? The set $A \Delta B$ is defined as follows: $$ A \Delta B \colon= ( A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B). $$
My Attempt:
We first show the following: If $S$ and $T$ are any subsets of $X$ such that $S \subset T$, then $\mathrm{Int} S \subset \mathrm{Int} T$.
So suppose that $S \subset T \subset X$. Let $x$ be any interior point of $S$. Then there exists an open set $U_x$ in $X$ such that $x \in U_x \subset S$. But $S \subset T$. So we can conclude that $U_x \subset T$, which implies that this $x$ is an interior point of $T$ also.
We also note that $\mathrm{Int} \emptyset = \emptyset$, because the only open set contained in $\emptyset$ is itself.
(i) Now as $A \subset A \cup B$ and $B \subset A \cup B$, so we must have $\mathrm{Int} A \subset \mathrm{Int} (A \cup B)$ and $\mathrm{Int} B \subset \mathrm{Int} (A \cup B)$, and hence $$\mathrm{Int} A \cup \mathrm{Int} B \subset \mathrm{Int} (A \cup B). \tag{0} $$ The reverse inclusion may not hold. For example, let $X = \mathbb{R}$, with the standard topology; let $A = \mathbb{Q}$ and $B = \mathbb{R} \setminus \mathbb{Q}$. Then $A \cup B = \mathbb{R}$ so that $\mathrm{Int} (A \cup B) = \mathbb{R}$, whereas $\mathrm{Int} A = \emptyset = \mathrm{Int} B$ and so $\mathrm{Int} A \cup \mathrm{Int} B = \emptyset$.
Under what most general possible condition(s) on $X$, if any, does the reverse inclusion also hold for every pair $A$, $B$ of subsets of $X$?
(ii) As $A \cap B \subset A$ and $A \cap B \subset B$, so we must have $\mathrm{Int} (A \cap B) \subset \mathrm{Int} A$ and $\mathrm{Int} (A \cap B) \subset \mathrm{Int} B$, which implies that $$\mathrm{Int} (A \cap B) \subset \mathrm{Int} A \cap \mathrm{Int} B. \tag{1} $$ Now let $x \in \mathrm{Int} A \cap \mathrm{Int} B$. Then $x \in \mathrm{Int} A$ and $x \in \mathrm{Int} B$, which implies that there are open sets $U_x$ and $V_x$ in $X$ such that $x \in U_x \subset A$ and $x \in V_x \subset B$; therefore $U_x \cap V_x$ is an open set in $X$ and $x \in U_x \cap V_x \subset A \cap B$, which implies that $x \in \mathrm{Int} (A \cap B)$. Thus we have shown that $$ \mathrm{Int} A \cap \mathrm{Int} B \subset \mathrm{Int} (A \cap B). \tag{2} $$ From (1) and (2) we can conclude that $$ \mathrm{Int} (A \cap B) = \mathrm{Int} A \cap \mathrm{Int} B. \tag{A} $$
Am I right?
(iii) As $A \setminus B \subset A$, so $\mathrm{Int} (A \setminus B) \subset \mathrm{Int} A$. Now let $x \in \mathrm{Int} (A \setminus B)$. Then $x \in \mathrm{Int} A$. We now show that this $x$ cannot be in $\mathrm{Int} B$. So suppose that $x \in \mathrm{Int} B$. Then by (A) above we can conclude that $$ x \in \mathrm{Int} A \cap \mathrm{Int} B = \mathrm{Int} (A \cap B). $$ But as $x \in \mathrm{Int} (A \setminus B)$, by our choice of $x$, so again by (A) above we can conclude that $$ x \in \mathrm{Int} (A \cap B) \cap \mathrm{Int} (A \setminus B) = \mathrm{Int} \left( (A \cap B) \cap (A \setminus B) \right) = \mathrm{Int} (\emptyset),$$ which is a contradiction, because $\mathrm{Int} (\emptyset) = \emptyset$. Thus our supposition that $x \in \mathrm{Int} (A \setminus B)$ and $x \in \mathrm{Int} B$ is wrong. So if $x \in \mathrm{Int} (A \setminus B)$, then $x \in \mathrm{Int} A$ but $x \not\in \mathrm{Int} B$, which implies that $x \in \mathrm{Int} A \setminus \mathrm{Int} B$. Therefore, we have $$ \mathrm{Int} (A \setminus B) \subset \mathrm{Int} A \setminus \mathrm{Int} B. \tag{3} $$ The reverse inclusion may not hold. For example, let $X = \mathbb{R}$, with the standard topology; let $A = [0, 2]$ and $B = [1, 3]$; then $\mathrm{Int} A = (0, 2)$ and $\mathrm{Int} B = (1, 3)$ so that $$\mathrm{Int} A \setminus \mathrm{Int} B = (0, 1]; $$ on the other hand, we note that $A \setminus B = [0, 1)$, and so $$ \mathrm{Int} (A \setminus B) = (0, 1) \subsetneqq (0, 1] = \mathrm{Int} A \setminus \mathrm{Int} B. $$
Am I right? Under what most general condition(s) on $X$, if any, does the reverse inclusion also hold for every pair of subsets $A$ and $B$ of $X$?
(iv) We note that $$ \begin{align} \mathrm{Int} A \Delta \mathrm{Int} B &= \left( \mathrm{Int} A \setminus \mathrm{Int} B \right) \cup \left( \mathrm{Int} B \setminus \mathrm{Int} A \right) \\ &\subset \mathrm{Int} \left( ( A \setminus B) \cup (B \setminus A) \right) \qquad \mbox{ [ using (0) above ] } \\ &= \mathrm{Int} ( A \Delta B). \tag{4} \end{align} $$ The reverse inclusion may not hold, however. As an example, take the same topological space $X$ and the same sets $A$ and $B$ as in (iii) above. We note that $A \Delta B = [0, 1) \cup (2, 3]$ so that $\mathrm{Int} (A \Delta B) = (0, 1) \cup (2, 3)$; on the other hand, $$ \mathrm{Int} A \Delta \mathrm{Int} B = (0, 2) \Delta (1, 3) = (0, 1] \cup [2, 3) \supsetneqq (0, 1) \cup (2, 3) = \mathrm{Int} (A \Delta B). $$
Am I right? What is the most general condition(s) on $X$ for the reverse inclusion to hold for every pair $A$ and $B$ of subsets of $X$?
