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It is a consequence of Baire's theorem that a connected, locally connected complete space cannot be written $ X = \bigcup_{n \geq 1}\ F_n$ where the $F_n$ are nonempty, pairwise disjoint closed sets.

Does anyone know of a counter-example to this if we don't assume the space to be locally connected?

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    You use the word *complete*, but you don't say *metric* space. Do you mean a complete metric space?2012-01-06

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If $e_n$ are the standard unit vectors in $\ell^2$, let $F_j$ consist of line segments from $e_j$ to $(1/j) e_j + e_k$ for $1 \le k < j$ ($F_1$ is the single point $e_1$). Then the $F_j$ are disjoint, closed and connected, and their union is closed and connected.

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    Indeed! Thank you very much.2012-01-06