8

Motivation:

We know that every open set is a countable union of open intervals with rational endpoints and that every open interval is a countable union of closed intervals. Hence every open set is a countable union of closed intervals. It follows by De Morgan's laws that every closed set is a countable intersection of open sets.

I would like to ask if we can prove a stronger result that

every closed set is a countable intersection of open intervals.

Thank you for your help!

YuiTo Cheng
  • 4,705
Akira
  • 17,367

1 Answers1

10

The answer is no, since an intersection of intervals is also an interval. Thus, if a closed set were to be a countable intersection of open intervals, it would have to be a closed interval, but there are closed sets that are not intervals.

For example (as @AlbertoTakase mentions in the comments below), consider the set $\{0\} \cup \{1\}$. This is a closed set since finite subsets of $\Bbb{R}$ are closed, but it is clearly not a closed interval. Hence, it cannot be written as a countable intersection of open intervals.

  • (to add an example) Consider ${0}\cup{1}$ which is closed but not an interval. – Alberto Takase Feb 20 '19 at 09:15
  • @AlbertoTakase Yes, but that is not a countable intersection of open intervals. –  Feb 20 '19 at 09:16
  • 1
    I only wanted to give an example of the last remark in your answer. – Alberto Takase Feb 20 '19 at 09:18
  • Your answer is great! – Akira Feb 20 '19 at 09:18
  • That's not necessarily true for countable intersections. Any set of isolated points (which is necessarily countable) is a countable intersection of open intervals. For example, ${0} = \bigcap_{n \in \Bbb N} (-2^{-n}, 2^{-n})$. – Robert Shore Feb 20 '19 at 09:31
  • 1
    @RobertShore Not any set of isolated points can be a countable intersection of open intervals, only singletons. And singletons are also (trivially) closed intervals. –  Feb 20 '19 at 09:37
  • 1
    @LeAnhDung Glad to be of help :) –  Feb 20 '19 at 09:38
  • @AlbertoTakase Ah, I see. I'll pull your example up into the answer for clarity :) –  Feb 20 '19 at 09:38