$A$ and $B$ are closed sets such that both $A\cup B$ and $A\cap B$ are connected. It is required to show that $A$ and $B$ are both connected. Any suggestions on how to tackle this problem?
2026-03-30 14:08:11.1774879691
A connectedness question
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Let $A=U \cup V$ where $U$ and $V$ are disjoint non-empty closed sets. Then $A \cap B= (U\cap B) \cup (V\cap B)$. Since $A\cap B$ is connected this gives $U\cap B =\emptyset$ or $V\cap B =\emptyset$. Suppose $U\cap B =\emptyset$. Then $A \cup B=(V\cup B) \cup U$ which contradicts connectedness of $A\cup B$.