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Find a countable set that is also open

seacoast123

New member
Feb 9, 2014
1
Find a countable set that is also open or prove that one cannot exist
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
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.