Complete, Equivalent, Closed sets

  • #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?
 

Answers and Replies

  • #2
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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?
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".
 
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  • #3
Kolmogorov states that equivalents sets are those on which a one-to-one correspondence can be found.
 
  • #4
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Kolmogorov states that equivalents sets are those on which a one-to-one correspondence can be found.
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".
 
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  • #5
Since both the open and closed sets on the real interval 0,1 have the same cardinality
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?
 
  • #6
CRGreathouse
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Can you clarify that statement a little?
If a < b, then |[a, b]| = |(a, b)| = |(a, b]| = |[a, b)| where |...| is the cardinality.
 
  • #7
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Also, would making the requirement more strict, maybe saying if the sets A and B are isometric, then A closed implies B is closed?
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.
 
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  • #8
Nevermind, I have the answer...it works when you have a metric preserving homeomorphism between the sets.
 

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