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The title pretty much is the question (by "closed interval" I mean to include possibly "degenerate" closed intervals consisting of a single point). The plausibility argument (I wouldn't quite call it a "proof") that leads me to think the answer is "yes" goes something like this:

We know the open set $O$ is a countable union of disjoint open intervals, call them $o_i$. It seems the closure of $O$ can, in general, introduce two kinds of limit points not already members of $O$:

  • Any endpoint of an $o_i$ (thus each $o_i$ "becomes" a closed interval in $\overline{O}$)
  • A point which is some finite distance from every $o_i$, but is the limit to which some (infinite) subset of the $o_i$ converges. It seems plausible (though I don't have a proof) that there would be at most countably many such limit points.

By this reasoning, the closure of any open subset of $\mathbb{R}$ would be equal to some countable union of closed (possibly "degenerate") intervals.

Is this right? Or am I missing some case where it fails to hold?

NikS
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    From How to ask a good question: "Your question should be clear without the title. After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title." – jjagmath Sep 04 '23 at 03:31

1 Answers1

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Great question! Perhaps surprisingly, the answer is negative: let $U$ be the open set consisting of an open interval of length $d\over 3$ placed in the center of each "deleted interval" in the construction of the Cantor set of length $d$. The closure of $U$ contains, in addition to the obvious countable collection of closed intervals, every point in the Cantor set itself.

(This construction is closely related to this answer to an old question of mine, incidentally.)

Noah Schweber
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  • Ah, yes. The Cantor set is starting to seem like the universal counter-example to every seemingly plausible but wrong intuition about the topology of $\mathbb{R}$. Thanks. – NikS Sep 05 '23 at 00:34