Let $X$ be a locally compact Hausdorff space and suppose that $X$ can be written as the disjoint union of countably many non-empty closed subsets. Is at least one of the subsets clopen?



closed decomposition of locally compact Hausdorff space
Let $X$ be a countable compact Hausdorff space consisting of a unique accumulation point $0$ with discrete complement $X'$. Let $j$ be a bijection $\mathbf{N}\to X'\times\mathbf{N}$. 


For $n\in\mathbf{N}$, write $F\_n=\{(0,n),j(n)\}$. This is a 2-element subset of $X\times\mathbf{N}$. Then $(F\_n)\_{n\in\mathbf{N}}$ is a countable closed partition of the Hausdorff locally compact space $X\times\mathbf{N}$, and none of the $F\_n$ is clopen.


(If we 1-point compactify, and add $F\_\infty=\{\infty\}$, we even have a compact example, homeomorphic to $X^2$.)


