Math 2.0

General topology Definition

Completion of a metric space

Builds on Complete metric spaceIsometryDense set

Definition

Let (X,d)(X, d) be a metric space. A completion of XX is a pair (X^,ι)(\hat{X}, \iota) of a complete metric space X^\hat{X} and an isometry ι ⁣:X→X^\iota \colon X \to \hat{X} with dense image ι(X)\iota(X).

Remarks

Each metric space has a completion. One construction takes the Cauchy sequences of XX and identifies two of them when the distance between their terms tends to 00. The completion is unique: for two completions (X^,ι)(\hat{X}, \iota) and (X^′,ι′)(\hat{X}', \iota'), there is a unique isometric isomorphism φ ⁣:X^→X^′\varphi \colon \hat{X} \to \hat{X}' with φ∘ι=ι′\varphi \circ \iota = \iota'. The completion of ℚ\mathbb{Q} with the usual metric is ℝ\mathbb{R}. This is the construction of the reals from Cauchy sequences. If XX is already complete, then XX with the identity map is a completion of XX.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

  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. Intersection of a family of sets definition
  22. Union of a family of sets definition
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Topological space definition
  27. Complement definition
  28. Function definition
  29. Metric space definition
  30. Convergence of a sequence in a metric space definition
  31. Isometry definition
  32. Open ball definition
  33. Diameter of a set definition
  34. Metric topology definition
  35. Epsilon characterisation of Cauchy sequences statement
  36. Complete metric space definition
  37. Open set definition
  38. Closed set definition
  39. Closure definition
  40. Dense set definition
  41. Completion of a metric space definition

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Complete_metric_space, Completion
About this page
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Kinddefinition
WritingW0 stub
MSC54E50
DefinesCompletion of a metric space
UsesComplete metric space, Isometry, Dense set

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