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Suppose for a compact topological space(if you want, we can assume Hausdorff) $X$, $X$ is a disjoint union of compact subsets,that is $X=\bigcup_{i\in I} X_i$ such that $X_i\cap X_j=\emptyset$ for $i\neq j$ and $X_i$ is compact for any $i\in I$. For convenience, call it a compact disjoint cover. Is it true that we can find a finite subcover of $X$ for any compact disjoint cover?

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    So $[0,1]=\bigcup_{x\in [0,1]} \{x\}$ would be a compact disjoint cover, and it obviously doesn't have a finite subcover. Is this really what you mean?2011-09-18

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