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Let $V=\cup_n I_n$ be a countable union of intervals in $\mathbb{R}$. Is the set of boundary points of $V$ countable? What if the intervals are strictly open (does this even make a difference)?

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No. Every open subset of $\mathbb{R}$ is the countable union of (strictly) open intervals (you can make them disjoint if you want). The complement of the Cantor set has the Cantor set as boundary, which is uncountable.

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    On this matter of points in the Cantor set that are not endpoints of any of the complements of deleted middle thirds, not that some of them are rational numbers: in particular $1/4$ and $3/10$ are members of the Cantor set.2012-09-28