Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
It is closed and has no interior points. That's all. – Daniel Fischer Aug 4 '14 at 21:59
Hint: Any open subset of $\mathbb{R}$ is a countable union of disjoint open intervals. – Random Jack Aug 4 '14 at 22:00
@DanielFischer is $\partial \mathbb{R}$ defined? – Stephen Nand-Lal Aug 4 '14 at 22:01
@StephenNand-Lal $\partial\mathbb{R} = \varnothing$. – Daniel Fischer Aug 4 '14 at 22:01
@RandomJack But the end points of intervals may have accumulation points which are not end points of intervals. – Per Manne Aug 4 '14 at 22:04

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.