Math 2.0

General topology Definition

Complete metric space

Builds on Metric spaceEpsilon characterisation of Cauchy sequencesConvergence of a sequence in a metric space

Definition

A metric space (X,d)(X, d) is complete when each Cauchy sequence in XX converges to a point of XX.

Remarks

The real line ℝ\mathbb{R} with the usual metric is complete. So is ℝn\mathbb{R}^n with the Euclidean metric. The rationals ℚ\mathbb{Q} and the open interval (0,1)(0, 1) are not complete. In (0,1)(0, 1), the sequence 1/n1/n is Cauchy but has no limit in (0,1)(0, 1). Completeness depends on the metric, not only on the topology: (0,1)(0, 1) is homeomorphic to ℝ\mathbb{R}, but only ℝ\mathbb{R} is complete with the usual metric. Each metric space has a completion.

Reading path

To understand this page from the ground up, read these 22 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 levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionEpsilon characterisation of Cauchy sequences, statement, no evidence levelEpsilon characterisat…statementComplete metric space, definition, no evidence levelComplete metric spacedefinition
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. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Supremum definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Metric space definition
  19. Convergence of a sequence in a metric space definition
  20. Diameter of a set definition
  21. Epsilon characterisation of Cauchy sequences statement
  22. Complete metric space definition

Where this leads

86 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Complete_metric_space, Definition
About this page
Tag08EWT7J0SC
URIm2://08EWT7J0SC
Kinddefinition
WritingW1 stated
MSC54E50
DefinesComplete metric space
UsesMetric space, Epsilon characterisation of Cauchy sequences, Convergence of a sequence in a metric space

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