consider X=(-1,0) union (0,1) with the metric induced from R . Then {(-1,0),(0,1)} is an open cover for X. show that this cover has no Lebesgue number

The Lebesgue number of an open cover of a metric space is a positive number such that every subset of the space with diameter less than this number is contained in some member of the cover.

In this case, we have the open cover {(-1,0),(0,1)} of the set X=(-1,0) union (0,1).

To show that this cover has no Lebesgue number, we need to find a subset of X with diameter less than any given positive number that is not contained in any member of the cover.

Consider the subset of X that is the interval (-ε/2, ε/2) for some positive ε. The diameter of this subset is ε, which can be made less than any given positive number by choosing ε sufficiently small.

However, this subset is not contained in any member of the cover. It is not contained in (-1,0) because it contains positive numbers, and it is not contained in (0,1) because it contains negative numbers.

Therefore, the cover {(-1,0),(0,1)} has no Lebesgue number.