Complete, Equivalent, Closed sets

In summary, two sets are considered equivalent if they have the same elements. However, equivalence does not necessarily mean that the sets have the same closure. The only inherent measure of a set is its cardinality, but if we have a metric preserving homeomorphism between sets A and B, then A being closed implies that B is also closed.
  • #1
Somefantastik
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0
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?
 
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  • #2
Somefantastik said:
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
Somefantastik said:
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
SW VandeCarr said:
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
Somefantastik said:
Can you clarify that statement a little?

If a < b, then |[a, b]| = |(a, b)| = |(a, b]| = |[a, b)| where |...| is the cardinality.
 
  • #7
Somefantastik said:
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.
 

What is a complete set?

A complete set is a set of elements that contains all possible members of a given group or collection. This means that every element in the group or collection is included in the set.

What is an equivalent set?

An equivalent set is a set that has the same number of elements as another set. This means that the two sets can be paired up in a one-to-one correspondence, with each element in one set corresponding to exactly one element in the other set.

What is a closed set?

A closed set is a set that contains all of its limit points. In other words, if a sequence of points within the set converges to a point outside of the set, then that point must also be included in the set.

How are complete, equivalent, and closed sets related?

Complete, equivalent, and closed sets are all different ways of describing sets that contain all of their elements. A complete set contains all possible elements, an equivalent set has the same number of elements as another set, and a closed set contains all of its limit points. They are all different ways of ensuring that a set is "complete" in some sense.

What are some examples of complete, equivalent, and closed sets?

An example of a complete set is the set of all real numbers, since it contains all possible real numbers. An example of an equivalent set is the set of even numbers and the set of integers, since they both have an infinite number of elements but can be paired up in a one-to-one correspondence. An example of a closed set is the closed interval [0,1], since it contains all of its limit points (such as 0 and 1).

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