Math 2.0

General topology Definition

Lebesgue covering dimension

Builds on Topological spaceOpen coverLebesgue number lemma

Definition

Let XX be a topological space. An open cover 𝒱\mathcal{V} of XX refines an open cover 𝒰\mathcal{U} if each member of 𝒱\mathcal{V} lies in some member of 𝒰\mathcal{U}. The order of a family 𝒱\mathcal{V} is the largest kk such that some point of XX lies in k+1k + 1 members of 𝒱\mathcal{V}. For an integer n≥−1n \ge -1, write dim⁡X≤n\dim X \le n if every finite open cover of XX has a finite open refinement of order at most nn. The Lebesgue covering dimension dim⁡X\dim X is the least nn with dim⁡X≤n\dim X \le n, and dim⁡X=∞\dim X = \infty if there is no such nn.

Remarks

The empty space has dim⁡∅=−1\dim \emptyset = -1, and a nonempty finite discrete space has dimension 00. The interval [0,1][0, 1] has dim⁡[0,1]=1\dim [0, 1] = 1. Given a finite open cover, the Lebesgue number lemma gives a number δ>0\delta > 0. Refine the cover by finitely many relatively open intervals of length less than δ\delta in which only neighbours overlap. This refinement has order at most 11. The cover {[0,2/3),(1/3,1]}\set{[0, 2/3), (1/3, 1]} has no open refinement of order 00. Such a refinement would be a cover by disjoint open sets, and by connectedness one of them would be all of [0,1][0, 1]. But [0,1][0, 1] lies in no member of the cover. So dim⁡[0,1]≤0\dim [0, 1] \le 0 fails. Some authors use all open covers instead of finite ones; the two versions agree for compact spaces. For every nn, dim⁡ℝn=n\dim \R^n = n.

Reading path

To understand this page from the ground up, read these 36 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 …objectOpen cover, concept, no evidence levelOpen coverconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionTopological space, concept, no evidence levelTopological spaceconceptSet, 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 lemmastatementLebesgue covering dimension, definition, no evidence levelLebesgue covering dim…definition
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.
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  1. Element definition
  2. Extended real number line object
  3. Open cover concept
  4. Proposition concept
  5. Real numbers object
  6. Set bounded above definition
  7. Topological space concept
  8. Set definition
  9. Conjunction definition
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Supremum definition
  14. Empty set object
  15. Family of sets definition
  16. Ordered pair definition
  17. Intersection definition
  18. Union definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Subset definition
  23. Union of a family of sets definition
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Topological space definition
  27. Function definition
  28. Metric space definition
  29. Open ball definition
  30. Diameter of a set definition
  31. Metric topology definition
  32. Open set definition
  33. Open cover definition
  34. Compact space definition
  35. Lebesgue number lemma statement
  36. Lebesgue covering dimension definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Lebesgue_covering_dimension
About this page
Tag99GNAS98PV
URIm2://99GNAS98PV
Kinddefinition
WritingW0 stub
MSC54F45
DefinesLebesgue covering dimension
UsesTopological space, Open cover, Lebesgue number lemma
Changed2 November 2026

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