Suppose $X$ is dense in $\mathbb R$ and $A\subset X$,then is it possible that $A$ and $X-A$ are both non-dense in $\mathbb R$.I think that at least one of them must be dense but I am having some problem to prove it.
2026-03-29 03:35:57.1774755357
Can a dense set $X$ in $\mathbb R$ contain a proper subset $A$ such that $A$ and $X-A$ are not dense in $\mathbb R$?
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How about $X=\mathbb R, A=(0,1)$?