Math 2.0

General topology Compactness Lemma

Lebesgue number lemma

Builds on Compact spaceOpen coverDiameter of a set

Statement

Let (X,d)(X, d) be a compact metric space and let (Ui)i∈I(U_i)_{i \in I} be an open cover of XX. Then there is a real δ>0\delta > 0 such that each subset A⊆XA \subseteq X with diameter diam⁡(A)<δ\operatorname{diam}(A) < \delta is contained in UiU_i for some i∈Ii \in I.

Remarks

Such a number δ\delta is a Lebesgue number of the cover. Equivalently, for each x∈Xx \in X, the ball B(x,δ/2)B(x, \delta / 2) lies in one of the sets UiU_i. Compactness is needed: the cover of (0,1)(0, 1) by the sets (1/n,1)(1/n, 1) has no Lebesgue number.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen cover, definition, no evidence levelOpen coverdefinitionCompact space, definition, no evidence levelCompact spacedefinitionLebesgue number lemma, statement, no evidence levelLebesgue number lemmastatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Supremum definition
  12. Empty set object
  13. Family of sets definition
  14. Ordered pair definition
  15. Intersection definition
  16. Union definition
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. Union of a family of sets definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Topological space definition
  25. Function definition
  26. Metric space definition
  27. Open ball definition
  28. Diameter of a set definition
  29. Metric topology definition
  30. Open set definition
  31. Open cover definition
  32. Compact space definition
  33. Lebesgue number lemma statement

Where this leads

20 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lebesgue%27s_number_lemma, lead section
About this page
Tag040CFNJEAP
URIm2://040CFNJEAP
Aliaseslebesgues-number-lemma
Kindstatement, lemma
WritingW0 stub
MSC54E45
StatusThis copy of the site has no build data. Run m2 status 040CFNJEAP.
UsesCompact space, Open cover, Diameter of a set
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