Unformatted text preview: covers X entirely. So let δ = min { ± ( x 1 ) / 2 ,...± ( x n ) / 2 } . Pick any x ∈ X , I then claim that B δ ( x ) ⊂ U for some U ⊂ U . Since x ∈ B ± ( x j ) / 2 ( x j ) for some x j , we take any y ∈ B δ ( x ), by triangular inequality: d ( y,x j ) ≤ d ( y,x ) + d ( x,x j ) < δ + d ( x,x j ) ≤ ± ( x j ). Hence y ∈ B ± ( x j ) ( x j ), this implies that B δ ( x ) ⊂ B ± ( x j ) ( x j ) ⊂ U x j 1...
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 Spring '12
 patirck
 Math, Topology, Metric space, xj, Lebesgue Covering Lemma

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