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Show disjoint sets are mutually separated?

  1. Sep 24, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that any two disjoint nonempty open sets are mutually separated.

    Show that any two disjoint nonempty closed sets are mutually separated.

    2. Relevant equations



    3. The attempt at a solution

    Let A and B be two disjoint nonempty open sets:

    Assume A and B share a boundary point.

    That boundary point exists in A intersect B. If the sets are disjoint, they can't intersect so this is a contradiction.

    Similar idea for closed sets.

    Is this correct?
     
  2. jcsd
  3. Sep 24, 2009 #2

    LCKurtz

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    No. What if A = (0,1) and B = (1,2). They both have 1 as a boundary point but it isn't in either.
     
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