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Homework Help: Analysis - Help me understand this dfn

  1. Jun 17, 2007 #1

    quasar987

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    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) [itex]\subset[/itex] 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) [itex]\subset[/itex] U_i for some i, then if r > a > 0, B(y,a) [itex]\subset[/itex] B(y,r) [itex]\subset[/itex] U_i, so that a is also a Lebesgue number.

    :grumpy:
     
  2. jcsd
  3. Jun 17, 2007 #2

    StatusX

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    Are you sure its not the supremum?
     
  4. Jun 17, 2007 #3

    quasar987

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

    Anyone know for sure?
     
  5. Jun 17, 2007 #4

    morphism

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    Another vote for supremum.

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