Why does the Lebesgue number not exist in a open and bounded set?

Answer 1

The Lebesgue number does not exist in an open and bounded set because such a set can contain arbitrarily small open balls around each of its points, making it impossible to find a single Lebesgue number that uniformly bounds the diameter of all the balls. In other words, for any positive number ε, there will always be points in the set that are closer together than ε, preventing the existence of a Lebesgue number. This property is a key reason why the Lebesgue number is not defined for open and bounded sets.

Answer 2

Because there are too many possibilities for the open covering. For a compact set, any family of open sets that covers can be replaced by a finite subfamily of open sets that still covers. Hence the open sets can't be too small.

Without compactness, the open sets can be quite small. For example, the infinite family of intervals (1/(n+1), 1/(n-1)) covers the open bounded set (0,1). (Here n is any integer larger than 1.) No subfamily will cover (0,1), and since the sets have radius going to zero, the Lebesgue number would also have to be zero.