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It is known that by Sierpinski: If a continuum $X$ has a countable cover $\{X_n\}$ by pairwise disjoint closed subsets, then at most one of sets $X_i$ is non-empty. A proof is given by the first answer in this link: Is $[0,1]$ a countable disjoint union of closed sets? I can understand the proof of lemma 1 and lemma 2. But I just can not figure out how by lemma 2, we can construct a decreasing sequence $C_n$. From the proof, it seems obvious, but I can not get the idea on its construction. Can any one help me?

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This question is simple after I find that we can use lemma 2 iteratively. Set $X=C_1$ and use the lemma 2 again , we get $C_2$, and set $X=C_2$ again. This gives the sequence clearly.

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