Complete space as a disjoint countable union of closed sets

Question

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?

Answer 1

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.