Let \(X\) be a connected space and \(A,B\subseteq X\) closed subsets with \(X=A\cup B\) and \(A\cap B\) connected. Show that \(A\) and \(B\) are connected.
Solution
Goal. For connected \(X = A \cup B\) with \(A, B\) closed and \(A \cap B\) connected, show \(A\) and \(B\) are connected.
1. Suppose \(A\) is disconnected.
assume for contradiction.
2. Then \(A = C \cup D\) with \(C, D\) nonempty, disjoint, and closed in \(A\) (hence closed in \(X\), since \(A\) is closed).
a disconnected space is a union of two nonempty disjoint closed subsets.
3. \(A \cap B\) is connected, so it lies entirely in \(C\) or entirely in \(D\).
\(A \cap B = (C \cap B) \cup (D \cap B)\) is a union of two disjoint closed sets; since \(A \cap B\) is connected, one of them is empty, so \(A \cap B \subseteq C\) or \(A \cap B \subseteq D\).
4. WLOG \(A \cap B \subseteq C\).
relabel if necessary.
5. Then \(X = (C \cup B) \cup D\) is a union of two disjoint nonempty closed sets.
5.1. \(C \cup B\) and \(D\) are disjoint.
5.2. \(C \cup B\) and \(D\) are closed.
\(C, D\) are closed in \(X\), and \(B\) is closed, so \(C \cup B\) is closed.
5.3. Both are nonempty.
\(C \neq \emptyset\) and \(D \neq \emptyset\).
5.4. \(X = (C \cup B) \cup D\).
\(X = A \cup B = (C \cup D) \cup B = (C \cup B) \cup D\).
6. This contradicts \(X\) being connected.
\(X\) is a union of two disjoint nonempty closed sets.
7. Hence \(A\) is connected; by symmetry, \(B\) is connected.
the same argument with \(A\) and \(B\) swapped.
8. Q.E.D.
7 is the claim.