Thread: Lebesgue's Number Lemma

Hello there! I have a fundamental (I suppose) question on Analysis. What is the Lebesgue's Number Lemma? What is all about? I read it but I can't figure out what it practically means. Could anybody explain to me?

Originally Posted by russel

Hello there! I have a fundamental (I suppose) question on Analysis. What is the Lebesgue's Number Lemma? What is all about? I read it but I can't figure out what it practically means. Could anybody explain to me?

Presumably this is the topological Lebesgue number lemma. Roughly, say that a metric space $\displaystyle X$ has the Lebesgue number property if for every open cover $\displaystyle \Omega$ of $\displaystyle X$ there exists some $\displaystyle \delta>0$ such that whenenver $\displaystyle E\subseteq \Omega$ satisfies $\displaystyle \text{diam}(\Omega)<\delta$ we must have that $\displaystyle E\subseteq O$ for some $\displaystyle O\in\Omega$. Then the Lebesgue number lemma states that every compact metric space has the Lebesgue number property. Roughly, this expresses the idea that for a compact metric space open covers have to cover 'a lot'. It is useful in proving the equivalence of compactness, limit point compactness, and sequential compactness in metric spaces as well as proving the Heine-Cantor theorem.