Let \(X\) be a topological space and \(U,V\subseteq X\) are two open disjoint subsets. If a subset \(A\subseteq X\) is connected and contained in \(U\cup V\) then either \(A\subseteq U\) or \(A\subseteq V\).
Proof
The subsets \(A\cap U\) and \(A\cap V\) disconnect \(A\) if both are non-empty.