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From Planetmath:

Lebesgue number lemma: For every open cover $\mathcal{U}$ of a compact metric space $X$, there exists a real number $\delta > 0$ such that every open ball in $X$ of radius $\delta$ is contained in some element of $\mathcal{U}$.

Any number $\delta$ satisfying the property above is called a Lebesgue number for the covering $\mathcal{U}$ in $X$.

I feel hard to picture and understand the significance of this result. I was wondering if there are some explanation for this lemma? Intuitively,

  1. a number bigger or smaller than a Lebesgue number may not be a Lebesgue number. So is a Lebesgue number simultaneously measuring how separated open subsets in an open cover are between each other, and how big each of them is?
  2. how is a metric space being compact make the existence of a Lebesgue number possible?
  3. Added: Is the lemma equivalent to say that for any open cover, there exist a positive number $\delta$, s.t. any open cover consisting of open balls with radius $\delta$ is always a refinement of the original open cover?

Thanks and regards!

  • 1
    http://planetmath.org/encyclopedia/ProofOfLebesgueNumberLemma.html2012-02-03
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    +1 because I also found it hard to picture and understand the significance. I don't remember much about it now, except that it was used in the proof of some other result not long after.2012-02-03
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    Any number _smaller_ than the Lebesgue number for a particular cover is also a Lebesgue number for that cover.2012-02-03
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    As a sidenote, it is used in one of the proofs of Seifert-van Kampen Theorem. For example, the one in Lee's *Topological Manifolds*.2012-02-03
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    And it is equivalent to the fact that the cover of all balls of radius $\delta$ is a refinement fo the original open cover. Lemma 2.27 here: https://arxiv.org/abs/1511.02057v22017-06-28

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