A Lebesgue number is a positive real number that represents the largest possible radius of small open balls that can be used to cover a given set without any overlap.
To prove that there is no Lebesgue number for a given set means to show that no matter how small the radius of the open balls used to cover the set is, there will always be some points in the set that are not covered by any ball.
To prove that there is no Lebesgue number for a specific open cover, you would need to show that for any given positive real number, there exists at least one point in the set that is not covered by any open ball with that radius.
Proving that there is no Lebesgue number for an open cover is important because it means that the set cannot be covered by a finite number of open balls, which has implications in various areas of mathematics and science.
No, it is not possible to prove that there is no Lebesgue number for any open cover. This is because there are some sets that do have a Lebesgue number, and it is only necessary to prove the non-existence of a Lebesgue number for a specific set and open cover, not for all sets and covers.