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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3It is closed and has no interior points. That's all. – Daniel Fischer Aug 04 '14 at 21:59
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3Hint: Any open subset of $\mathbb{R}$ is a countable union of disjoint open intervals. – Random Jack Aug 04 '14 at 22:00
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@DanielFischer is $\partial \mathbb{R}$ defined? – Stephen Nand-Lal Aug 04 '14 at 22:01
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2@StephenNand-Lal $\partial\mathbb{R} = \varnothing$. – Daniel Fischer Aug 04 '14 at 22:01
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What is the boundary of the complement of the Cantor set? Is it countable? – Somabha Mukherjee Aug 04 '14 at 22:03
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A full characterization is in Daniel Fischer's comment. – Jonas Meyer Aug 04 '14 at 22:03
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@SomabhaMukherjee: It is the Cantor set. – Jonas Meyer Aug 04 '14 at 22:03
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2@RandomJack But the end points of intervals may have accumulation points which are not end points of intervals. – Per Erik Manne Aug 04 '14 at 22:04
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So, it is uncountable, but of Lebesgue measure 0. Can anyone give an example, where the boundary of an open set has non-zero Lebesgue measure? – Somabha Mukherjee Aug 04 '14 at 22:05
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You should consider the boundary of the complement of a fat Cantor set as well. – David Mitra Aug 04 '14 at 22:05
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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.
Daniel Fischer
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