Find a countable set that is also open

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No countable subset of the real line can be open. The proof begins by assuming that a countable open subset \( C \) exists within \( \mathbb{R} \) and selecting any point \( x \) in \( C \). For this point, there exists a \( \delta > 0 \) such that the interval \( (x-\delta, x+\delta) \) is entirely contained in \( C \). However, this interval is uncountable, leading to the conclusion that \( C \) cannot be countable.

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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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