Let $A$ and $B$ be two closed subsets of a topological space $X$ such that both $A \cup B$ and $A \cap B$ are connected. Prove that both $A$ and $B$ are also connected.
My attempt:
On the contrary, let us assume that $A$ is disconnected. Then there exist non-empty disjoint closed subsets $C$ and $D$ of $A$ such that $A = C \cup D.$
Since $A$ is a closed subset of $X$ so are $C$ and $D$. Then both $C \cap B$ and $D \cap B$ are disjoint closed subsets of $A \cap B$. As $A \cap B$ is connected, so at least one of $C \cap B$ or $D \cap B$ is empty.
Now if $C \cap B = \emptyset$, then $$ A \cup B = (D \cup B) \cup C $$ gives a disconnection of $A \cup B$, and if $D \cap B = \emptyset$, then $$ A \cup B = (C \cup B) \cup D$$ gives a disconnection of $A \cup B$. So in any case we arrive at a contradiction. Hence $A$ has to be connected.
By a similar argument we can prove that $B$ has to be connected and this completes the proof.
Now my question is "What will happen if we drop closedness of at least one of the sets $A$ or $B$?" Does the result still hold?
Any help in this regard will be highly appreciated. Also please check my proof whether it holds good or not.
Thanks in advance.
My only issue with your attempt is "Since $A$ is a closed subset of $X$ so are $C$ and $D$". This statement is not needed since $C, D$ are closed by assumption.
Regarding your other question, consider $(\mathbb R, \tau)$ with $A = (0,1)\cup (2,3)$, $B=[1,2]$. Then $A\cup B = (0,3)$ is connected, so is $A\cap B = \emptyset$ (vacuously) but $A$ is disconnected.