Let $\Omega\subset (0,1)$ be a nowhere dense set which has no lower and upper bound in $(0,1)$ and for which $\Omega^{d}\cap(0,1)=\Omega$ ($\Omega^{d}$ denotes here the set of all limit points of $\Omega$; let also $\Omega^{\pm d}$ denote the set of all two-sided limit points of $\Omega$). Is it true that then the set $(0,1)\backslash \Omega^{\pm d}$ is the set which is the union of the disjoint closed intervals $[a,b]$, where $a$ and $b$ are left-sided and right-sided limit points of $\Omega$, respectively? I think it's true but have no idea how to proceed with the proof. Thank you for any replies.
Question related to some class of nowhere dense sets
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general-topology
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0Now I see your point. Thank you for your tips. They were very helpful. – 2012-06-02