Metric and Topological Spaces | Part IB, 2005

Let XX be a topological space. Suppose that U1,U2,U_{1}, U_{2}, \ldots are connected subsets of XX with UjU1U_{j} \cap U_{1} non-empty for all j>0j>0. Prove that

W=j>0UjW=\bigcup_{j>0} U_{j}

is connected. If each UjU_{j} is path-connected, prove that WW is path-connected.

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