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How to see that every uncountable closed subset $A$ of $\mathbb{R}$ is the union of countably many compact subsets of $\mathbb{R}$?

Thanks ahead.

Dave L. Renfro
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1 Answers1

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For each $n\ge 1$, let $C_n=[-n,n]\cap A$. Then $C_n$ is closed since it an intersection of two closed sets. It is clearly bounded. So it is compact. But $A=\bigcup_n C_n$, proving the claim. (the information that $A$ is uncountable is redundant.)

Ittay Weiss
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