Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is a non-open, countable intersection of open sets closed?

share|improve this question
What is a non-open intersection? –  Git Gud Mar 12 '13 at 20:45
To clarify, you are asking 'If the countable intersection of open sets is not open, does this imply that the intersection is closed?' –  Daniel Rust Mar 12 '13 at 20:45
Yes, that's correct. –  Cleaner Mar 12 '13 at 20:45
@GitGud: Suppose that $\mathscr{U}$ is a family of open sets such that $\bigcap\mathscr{U}$ is not open. The OP is asking whether it is then the case that $\bigcap\mathscr{U}$ must be closed. –  Brian M. Scott Mar 12 '13 at 20:49
@Cleaner I see this is your fourth question here and you haven't accepted any answers so far. Please consider going through your questions and accepting answers to the question in wich you got at least a satisfying answer. –  Git Gud Mar 12 '13 at 20:55

3 Answers 3

up vote 8 down vote accepted

Not necessarily: in $\Bbb R$ take the intersection of the sets $\left(-\frac1n,1\right)$ for $n\in\Bbb Z^+$: you get $[0,1)$.

share|improve this answer

Maybe. A countable intersection of open sets is called a G$_\delta$-set. Examples of these (in the real line) include

  • $[ 0 , 1 ] = \bigcap_{n} ( - \frac{1}{n} , 1+ \frac{1}{n} )$;
  • $( 0 , 1 ] = \bigcap_{n} ( 0 , 1+ \frac{1}{n} )$;
  • the set $\mathbb{R} \setminus \mathbb{Q}$ of all irrational numbers: $\bigcap_{q \in \mathbb{Q}} ( \mathbb{R} \setminus \{ q \} )$.
share|improve this answer

Consider the intersection of the open intervals $\left(0, 1+\frac{1}{n}\right)$, where $n$ ranges over the positive integers.

share|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.