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Complete, Equivalent, Closed sets

  1. Mar 7, 2010 #1
    If a set A and a set B are equivalent, and it is known that A is complete, can it then be said that B is also complete?

    What if it is known that A is closed, can it then be said that B is also closed?
     
  2. jcsd
  3. Mar 8, 2010 #2
    Two sets are equal if they have the same elements. So the set A=B={[0,1]} is closed because it contains both 0 and 1. I don't know if you mean something different by using the term "equivalent".
     
    Last edited: Mar 8, 2010
  4. Mar 8, 2010 #3
    Kolmogorov states that equivalents sets are those on which a one-to-one correspondence can be found.
     
  5. Mar 8, 2010 #4
    This means two sets have the same cardinality. Since both the open and closed sets on the real interval 0,1 have the same cardinality, A and B may nevertheless differ in terms of closure. If you wish to call the closed interval "complete" then the open interval would not be "complete".
     
    Last edited: Mar 8, 2010
  6. Mar 8, 2010 #5
    Can you clarify that statement a little?

    Also, would making the requirement more strict, maybe saying if the sets A and B are isometric, then A closed implies B is closed?
     
  7. Mar 8, 2010 #6

    CRGreathouse

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    If a < b, then |[a, b]| = |(a, b)| = |(a, b]| = |[a, b)| where |...| is the cardinality.
     
  8. Mar 8, 2010 #7
    What metric are you defining for your sets? The only inherent measure of a set is the number of elements it contains (ie its cardinality). The fact that sets A and B have the same cardinality doesn't imply that if A is closed, B must be closed.
     
    Last edited: Mar 8, 2010
  9. Mar 9, 2010 #8
    Nevermind, I have the answer...it works when you have a metric preserving homeomorphism between the sets.
     
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