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#### seacoast123

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- Feb 9, 2014

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Find a countable set that is also open or prove that one cannot exist

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- Feb 9, 2014

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Find a countable set that is also open or prove that one cannot exist

- Mar 10, 2012

- 835

No countable subset of the real line is open. To prove it, assume $C$ is a countable open subset of $\mathbb R$ and $x$ be any point in $C$.Find a countable set that is also open or prove that one cannot exist

Then there exists $\delta>0$ such that $(x-\delta,x+\delta)\subseteq C$.

But $(x-\delta,x+\delta)$ is uncountable (why?).

Hence $C$ cannot be countable.