MHB Find a countable set that is also open

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No countable subset of the real line can be open. The argument begins by assuming a countable open set C exists in the real numbers and selecting any point x from C. For x, there exists a delta such that the interval (x - delta, x + delta) is entirely contained within C. However, this interval is uncountable, leading to the conclusion that C cannot be countable. Thus, a countable open set in the real line cannot exist.
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Find a countable set that is also open or prove that one cannot exist
 
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seacoast123 said:
Find a countable set that is also open or prove that one cannot exist
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 $C$.

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.
 

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