Is Any Number Smaller Than a Lebesgue Number Also a Lebesgue Number?

[quasar987]

Homework Statement

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.

 

[StatusX]

Are you sure its not the supremum?

 

[quasar987]

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

Anyone know for sure?

 

[morphism]

Another vote for supremum.

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