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What if it is known that A is closed, can it then be said that B is also closed?

- Thread starter Somefantastik
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What if it is known that A is closed, can it then be said that B is also closed?

- #2

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Two sets are

What if it is known that A is closed, can it then be said that B is also closed?

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- #3

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

- #4

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

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Can you clarify that statement a little?Since both the open and closed sets on the real interval 0,1 have the same cardinality

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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If a < b, then |[a, b]| = |(a, b)| = |(a, b]| = |[a, b)| where |...| is the cardinality.Can you clarify that statement a little?

- #7

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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.Also, would making the requirement more strict, maybe saying if the sets A and B are isometric, then A closed implies B is closed?

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