Math 2.0

General topology Proposition

Complete subspaces are closed

Builds on Complete metric spaceClosure is the set of limits of sequencesSubspace metric

Statement

Let (X,d)(X, d) be a metric space and let A⊆XA \subseteq X. If AA with the subspace metric is complete, then AA is closed in XX.

Remarks

Proof: let x∈A‾x \in \overline{A}. By closure is the set of limits of sequences, there is a sequence in AA with limit xx. It is a Cauchy sequence in AA, so it has a limit in AA. Limits are unique in XX, so x∈Ax \in A. The space XX need not be complete here. For example, ℚ\mathbb{Q} is not closed in ℝ\mathbb{R}, so ℚ\mathbb{Q} is not complete. In a complete space, a subset is complete if and only if it is closed, by closed subsets of complete spaces are complete.

Reading path

To understand this page from the ground up, read these 45 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. Restriction of a function definition
  30. Metric space definition
  31. Subspace metric definition
  32. Convergence of a sequence in a metric space definition
  33. Open ball definition
  34. Diameter of a set definition
  35. Metric topology definition
  36. Epsilon characterisation of Cauchy sequences statement
  37. Complete metric space definition
  38. Open set definition
  39. Closed set definition
  40. Open sets of a metric space statement
  41. Closed sets contain their limits statement
  42. Closure definition
  43. Closure in a metric space statement
  44. Closure is the set of limits of sequences statement
  45. Complete subspaces are closed statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Complete_metric_space, Some theorems
About this page
Tag0W41XG4W60
URIm2://0W41XG4W60
Kindstatement, proposition
WritingW0 stub
MSC54E50
StatusThis copy of the site has no build data. Run m2 status 0W41XG4W60.
UsesComplete metric space, Closure is the set of limits of sequences, Subspace metric

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