Find a countable set that is also open

[seacoast123]

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

 

[caffeinemachine]

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.