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Can we say anything about the boundary of an open subset of the real numbers (in the usual topology generated by open intervals)? For example, it is countable, or has Lebesgue measure 0, etc?

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It is closed and has no interior points. That's all. –  Daniel Fischer Aug 4 at 21:59
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Hint: Any open subset of $\mathbb{R}$ is a countable union of disjoint open intervals. –  Random Jack Aug 4 at 22:00
    
@DanielFischer is $\partial \mathbb{R}$ defined? –  Stephen Nand-Lal Aug 4 at 22:01
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@StephenNand-Lal $\partial\mathbb{R} = \varnothing$. –  Daniel Fischer Aug 4 at 22:01
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@RandomJack But the end points of intervals may have accumulation points which are not end points of intervals. –  Per Manne Aug 4 at 22:04

1 Answer 1

Every boundary in any topological space is closed. The boundary of an open set has empty interior.

Every closed set with empty interior is the boundary of its complement.

Therefore, the family of boundaries of open subsets of $\mathbb{R}$ is the family of closed sets with empty interior. The classical cantor set is an example of an uncountable closed set with empty interior, and fat Cantor sets are examples of closed sets with empty interior and positive measure.

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