Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook