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Show that in $\mathbb R^2$ every closed set is boundary for some open set.

Please give hint .

I think it is also true for all $\mathbb R^n$ . Is this proof similar to the above ?

Stefan Hamcke
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bytrz
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    The boundary of an open set has empty interior. But not every closed set has empty interior. – Stefan Hamcke Apr 29 '14 at 21:17
  • Perhaps you meant every boundary set is closed? –  Apr 29 '14 at 21:34
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    Perhaps you meant the problem in this question? – Old John Apr 29 '14 at 21:36
  • @OldJohn actually i saw this question as your link but then i find that on Wikipedia (http://en.wikipedia.org/wiki/Boundary_%28topology%29) it is saying every closed set is boundary for some open set (at the properties part of wikipedia link ). Answers at your link find a set that is not open . – bytrz Apr 29 '14 at 22:11
  • @NotNotLogical No i don't mean it you can see my question at wikipedia link at the part of properties. – bytrz Apr 29 '14 at 22:13
  • @StefanHamcke good point thanks maybe it is written wrongly at en.wikipedia.org/wiki/Boundary_%28topology%29 ? – bytrz Apr 29 '14 at 22:38

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To summarise the above comments: it is not true that every closed set in $\mathbb{R}^2$ is the boundary of an open set. Such sets have empty interior, and many closed sets in the plane do have non-empty interior.

The elegant argument at this answer shows that any resolvable space $X$ (which means it can be written as the union of two disjoint dense subsets) has the property that every closed set is the boundary of some set. And all Euclidean spaces are resolvable ($\mathbb{Q}^n$ and its complement will do, e.g.).

So the linked statement at wikipedia is false, but something weaker is true, in a more general setting. [Added] The statement has now been reverted to a correct statement, see the link in the comment below!

Community
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Henno Brandsma
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    The statement on Wikipedia was simply "every closed set is the boundary of some set" until someone mistakenly added the qualifier "open": http://en.wikipedia.org/w/index.php?title=Boundary_(topology)&diff=573715413&oldid=555557870 – Samuel May 04 '14 at 12:13
  • @Samuel Thanks it explains everything :) – bytrz May 04 '14 at 15:31