Math 2.0

General topology Proposition

Closed subsets of complete spaces are complete

Builds on Complete metric spaceClosed sets contain their limitsSubspace metric

Statement

Let (X,d)(X, d) be a complete metric space and let FF be a closed subset of XX. Then FF with the subspace metric is complete.

Remarks

Proof: let (xn)(x_n) be a Cauchy sequence in FF. It is Cauchy in XX, so it converges to some x∈Xx \in X. Then x∈Fx \in F, because closed sets contain their limits. For example, [0,1][0, 1], ℤ\mathbb{Z} and each closed ball of ℝn\mathbb{R}^n are complete. The converse is complete subspaces are closed.

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. Union of a family of sets definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Universal set definition
  25. Topological space definition
  26. Complement definition
  27. Function definition
  28. Restriction of a function definition
  29. Metric space definition
  30. Subspace metric definition
  31. Convergence of a sequence in a metric space 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. Open sets of a metric space statement
  40. Closed sets contain their limits statement
  41. Closed subsets of complete spaces are complete statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Complete_metric_space, Some theorems
About this page
Tag0FBXJZX3MK
URIm2://0FBXJZX3MK
Kindstatement, proposition
WritingW0 stub
MSC54E50
StatusThis copy of the site has no build data. Run m2 status 0FBXJZX3MK.
UsesComplete metric space, Closed sets contain their limits, Subspace metric

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