Let \(\ts{A_\alpha}\) be a collection of connected subspaces of \(X\); let \(A\) be a connected subspace of \(X\). Show that if \(A \cap A_\alpha \neq \varnothing\) for all \(\alpha\), then \(A \cup (\bigcup A_\alpha)\) is connected.
Solution
Let \[ C=A\cup\bigcup_\alpha A_\alpha. \] Suppose \(C=U\cup V\) is a separation. Since \(A\) is connected, it lies entirely in one side, say \(A\subseteq U\). For each \(\alpha\), the connected set \(A_\alpha\) meets \(A\), hence meets \(U\). Therefore \(A_\alpha\) cannot meet \(V\), so \(A_\alpha\subseteq U\). Thus \(C\subseteq U\), contradicting that \(V\) is nonempty. Hence \(C\) is connected.