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The Boundary of a Countable Union of Almost Disjoint Cubes

  1. Sep 26, 2010 #1
    Let E be a subset of R2 that is non-empty, compact, and connected. Suppose furthermore that E is the union of a countably infinite number of almost disjoint closed cubes {Ri} with non-zero volume.

    Is there anything interesting about this set, particularly its boundary? Can it have infinite length, for example? I can't think of very much to say about it, at all.
  2. jcsd
  3. Sep 26, 2010 #2


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    What do you mean by almost disjoint? Their intersection has measure zero?
  4. Sep 26, 2010 #3
    My apologies, yes. If two cubes are almost disjoint then only their boundaries intersect.
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