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Lebesgue's number lemma

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In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states: Such a number 𝛿 is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well. Let 𝑈 be an open cover of 𝑋 . Since 𝑋 is compact we can extract a finite subcover { 𝐴 1 , … , 𝐴 𝑛 } ⊆ 𝑈 . If any one of the 𝐴 𝑖 's equals 𝑋 then any 𝛿 > 0 will serve as a Lebesgue number. Otherwise for each 𝑖 ∈ { 1 , … , 𝑛 } , let 𝐶 𝑖 := 𝑋 ∖ 𝐴 𝑖 , note that 𝐶 𝑖 is not empty, and define a function 𝑓 : 𝑋 → 𝑅 by Since 𝑓 is continuous on a compact set, it attains a minimum 𝛿 . The key observation is that, since every 𝑥 is contained in some 𝐴 𝑖 , the extreme value theorem shows 𝛿 > 0 . Now we can verify that this 𝛿 is the desired Lebesgue number. If 𝑌 is a subset of 𝑋 of diameter less than 𝛿 , then there exists 𝑥 0 ∈ 𝑋 such that 𝑌 ⊆ 𝐵 𝛿 ( 𝑥 0 ) , wh

Lebesgue's number lemma - Wikipedia Jump to content From Wikipedia, the free encyclopedia Given a cover of a compact metric space, all small subsets are subset of some cover set In topology , the Lebesgue covering lemma is a useful tool in the study of compact metric spaces . Given an open cover of a compact metric space X {\displaystyle X} , a Lebesgue's number of the cover is a number 0"}}'> 0}"> δ > 0 {\displaystyle \delta >0} 0}"/> such that every subset of X {\displaystyle X} having diameter less than δ {\displaystyle \delta } is contained in some member of the cover. The existence of Leb

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