Math 2.0

General topology Proof

Proof that closed subsets of complete spaces are complete

Builds on Complete metric spaceClosed setSubspace metricEpsilon characterisation of Cauchy sequencesConvergence of a sequence in a metric spaceClosed sets contain their limits

Proof

  1. Let (X,d)(X, d) be a complete metric space and let FF be a closed subset of XX. Let dFd_F be the subspace metric on FF.
  2. Let (xn)(x_n) be a Cauchy sequence in (F,dF)(F, d_F).
  3. We have d(xm,xn)=dF(xm,xn)d(x_m, x_n) = d_F(x_m, x_n) for all m,nm, n. So (xn)(x_n) is also a Cauchy sequence in (X,d)(X, d).
  4. Since XX is complete, (xn)(x_n) converges in XX to some point x∈Xx \in X.
  5. All xnx_n lie in FF, and FF is closed. Since closed sets contain their limits, we get x∈Fx \in F.
  6. For each nn, dF(xn,x)=d(xn,x)d_F(x_n, x) = d(x_n, x). So dF(xn,x)→0d_F(x_n, x) \to 0, and (xn)(x_n) converges to xx in (F,dF)(F, d_F).
  7. Thus each Cauchy sequence in FF converges to a point of FF, and (F,dF)(F, d_F) is complete.

Idea

A Cauchy sequence in FF has a limit in the big space, because the big space is complete. A closed set cannot lose a limit of its own points, so the limit lies in FF.

Reading path

To understand this page from the ground up, read these 44 pages in order. The foundations come first. The path takes the proofs Proof of the diameter characterisation of Cauchy sequences, Proof of the description of the open sets of a metric space, Proof that closed sets contain their limits.

  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. Proof of the diameter characterisation of Cauchy sequences proof
  36. Open set definition
  37. Epsilon characterisation of Cauchy sequences statement
  38. Closed set definition
  39. Proof of the description of the open sets of a metric space proof
  40. Complete metric space definition
  41. Open sets of a metric space statement
  42. Proof that closed sets contain their limits proof
  43. Closed sets contain their limits statement
  44. Proof that closed subsets of complete spaces are complete proof

Where this leads

59 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Subspace_of_Complete_Metric_Space_is_Closed_iff_Complete, Necessary Condition
About this page
Tag8ETHA6K7BY
URIm2://8ETHA6K7BY
Kindproof
WritingW0 stub
MSC54E50
ProvesClosed subsets of complete spaces are complete
UsesComplete metric space, Closed set, Subspace metric, Epsilon characterisation of Cauchy sequences, Convergence of a sequence in a metric space, Closed sets contain their limits
Changed4 November 2026

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