Math 2.0

General topology Proof

Proof that closed sets contain their limits

Builds on Metric spaceClosed setConvergence of a sequence in a metric spaceOpen sets of a metric spaceOpen ball

Proof

We argue by contradiction.

  1. Let (X,d)(X, d) be a metric space and let FF be a closed subset of XX.
  2. Let (xn)(x_n) be a sequence of points of FF that converges to x∈Xx \in X.
  3. Suppose that x∉Fx \notin F. Then x∈X∖Fx \in X \setminus F, and X∖FX \setminus F is open because FF is closed.
  4. By the description of the open sets of a metric space, there is ε>0\varepsilon > 0 with B(x,ε)⊆X∖FB(x, \varepsilon) \subseteq X \setminus F. Here B(x,ε)B(x, \varepsilon) is the open ball.
  5. Since xn→xx_n \to x, there is an integer NN with d(xn,x)<εd(x_n, x) < \varepsilon for all n≥Nn \ge N.
  6. By the symmetry of dd, d(x,xN)<εd(x, x_N) < \varepsilon, so xN∈B(x,ε)⊆X∖Fx_N \in B(x, \varepsilon) \subseteq X \setminus F.
  7. This contradicts xN∈Fx_N \in F. Hence x∈Fx \in F.

Idea

If the limit were outside FF, it would have a ball around it that misses FF. The terms of the sequence eventually enter every ball around the limit. They would then leave FF, which they never do.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first. The path takes the proof Proof of the description of the open sets of a metric space.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionProof of the description of the open sets of a metric space, proof, no evidence levelProof of the descript…proofClosed set, definition, no evidence levelClosed setdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementProof that closed sets contain their limits, proof, no evidence levelProof that closed set…proof
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.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Metric space definition
  26. Convergence of a sequence in a metric space definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Proof of the description of the open sets of a metric space proof
  31. Closed set definition
  32. Open sets of a metric space statement
  33. Proof that closed sets contain their limits proof

Where this leads

712 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closed_set
About this page
TagMGR5JY7H71
URIm2://MGR5JY7H71
Kindproof
WritingW0 stub
MSC54E35
ProvesClosed sets contain their limits
UsesMetric space, Closed set, Convergence of a sequence in a metric space, Open sets of a metric space, Open ball
Changed4 November 2026

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