Give an example of a compact countable infinite subset of $\mathbb{R}$. I'm having a difficult time, because I know that closed intervals $[a,b]$ are compact and infinite but are uncountable. Any help would be appreciated. Thanks!
Give an example of a compact countable infinite subset of $\mathbb{R}$
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real-analysis
general-topology
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4Do you know that a subset of $\mathbb{R}$ is compact if and only if it's closed and bounded? Can you find an example of a countable set which is closed and bounded? – 2011-05-12
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1Once one is aware of the example of a convergent sequence and its limit, an interesting follow-up is trying to understand what countable linear orders are possible as compact countable subsets of ${\mathbb R}$. – 2011-05-12
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0Thank you! So for instance an example would be: Let (sn)=1/n. Then it would be {1/n U 0}? – 2011-05-12
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0Yes, that would work. Is it clear to you why this is compact? – 2011-05-12
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0@Andres, is there a reasonable way to characterize such linear orders? – 2011-05-12
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1Yes it is closed because it contains all the limit points and bounded since it is bounded below by 0 and above by 1. Then any closed and bounded subset of R^n is compact. – 2011-05-12
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0@Alon: I know of partial results, but not a full answer. I think it is an interesting question. – 2011-05-12