# Complete, Equivalent, Closed sets

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?

Related Set Theory, Logic, Probability, Statistics News on Phys.org
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".

Last edited:
Kolmogorov states that equivalents sets are those on which a one-to-one correspondence can be found.

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".

Last edited:
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?

CRGreathouse