Let \(\ts{A_n}\) be a sequence of connected subspaces of \(X\), such that \(A_n \cap A_{n+1} \neq \varnothing\) for all \(n\). Show that \(\bigcup A_n\) is connected.
Solution
Set \[ B_n=A_1\cup\cdots\cup A_n. \] We prove inductively that \(B_n\) is connected. The case \(n=1\) is given. If \(B_n\) is connected, then \[ B_n\cap A_{n+1}\supseteq A_n\cap A_{n+1}\ne\varnothing, \] so the union \(B_{n+1}=B_n\cup A_{n+1}\) of two intersecting connected sets is connected.
The sets \(B_n\) form an increasing sequence of connected subspaces and all contain \(A_1\). Hence their union is connected. Since \[ \bigcup_nB_n=\bigcup_nA_n, \] the desired union is connected.