Suppose \(A, B\subseteq \RR^n\) are connected and not disjoint. Prove that \(A\union B\) is also connected.
Solution
Suppose, for contradiction, that \(A\cup B\) is disconnected. Then there are disjoint nonempty sets \(U,V\), open in the subspace topology of \(A\cup B\), such that \[ A\cup B=U\cup V. \] Because \(A\) is connected, it must lie entirely in one of \(U,V\); likewise \(B\) must lie entirely in one of them. Since \(A\cap B\ne\varnothing\), the two sets cannot lie in different members of the separation. Hence both \(A\) and \(B\) lie in the same one, leaving the other empty, a contradiction. Therefore \(A\cup B\) is connected.