# Analysis - Help me understand this dfn

1. Jun 17, 2007

### quasar987

1. The problem statement, all variables and given/known data
Let A be a set in a metric space and {U_i} be an open cover of A. A number r > 0 such that for all y in A, B(y,r) $\subset$ U_i for some i is called a Lebesgue number for the covering. The infimum of all Legesgue number is called the Lebesgue number for the covering.

Am I wrong in thinking that if r is a Lebesgue number, then any other number lesser than r is also a Lebesgue number, so that if soon as a Lebesgue number exists, the Lebesgue number for the covering is 0?

After all, if for some r > 0, B(y,r) $\subset$ U_i for some i, then if r > a > 0, B(y,a) $\subset$ B(y,r) $\subset$ U_i, so that a is also a Lebesgue number.

:grumpy:

2. Jun 17, 2007

### StatusX

Are you sure its not the supremum?

3. Jun 17, 2007

### quasar987

It's infimum in my book. And wiki doesn'T make a distinction btw a and the Lebesgue number.

Anyone know for sure?

4. Jun 17, 2007

### morphism

Another vote for supremum.

Although I haven't seen a definition for the Lebesgue number before.