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Open subset of a perfect set.

  1. Sep 18, 2013 #1
    Suppose we have a perfect set [itex]E\subset\mathbb{R}^k[/itex]. Is there an open set [itex]I\subset E[/itex]?
     
  2. jcsd
  3. Sep 18, 2013 #2

    mathman

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    I'm rusty at this. However, I understand that a closed interval is a perfect set. Take the closed unit cube in Rk, drop all boundary points leaving an open unit cube.
     
  4. Sep 18, 2013 #3

    Office_Shredder

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    Sometime's it's true (like mathman's example), and sometimes it's false. For example the Cantor set is a perfect set but contains no open interval inside of it
     
  5. Sep 19, 2013 #4
    Yeah, I just found out that. Thank you :)
     
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