Math 2.0

General topology Proof

Proof that closed balls are closed

Builds on Metric spaceClosed ballOpen ballTriangle inequality for a metricOpen sets of a metric spaceClosed set

Proof

We show that the complement of the closed ball is open.

  1. Let (X,d)(X, d) be a metric space, let x∈Xx \in X and let r≥0r \ge 0. Let Bˉ(x,r)={z∈X:d(x,z)≤r}\bar{B}(x, r) = \set{z \in X : d(x, z) \le r} be the closed ball.
  2. Put C=X∖Bˉ(x,r)C = X \setminus \bar{B}(x, r). Let y∈Cy \in C. Then d(x,y)>rd(x, y) > r.
  3. Put ε=d(x,y)−r\varepsilon = d(x, y) - r. By step 2, ε>0\varepsilon > 0.
  4. Let zz be in the open ball B(y,ε)B(y, \varepsilon), so d(y,z)<εd(y, z) < \varepsilon.
  5. By the triangle inequality and the symmetry of dd, d(x,y)≤d(x,z)+d(z,y)=d(x,z)+d(y,z)d(x, y) \le d(x, z) + d(z, y) = d(x, z) + d(y, z).
  6. Hence d(x,z)≥d(x,y)−d(y,z)>d(x,y)−ε=rd(x, z) \ge d(x, y) - d(y, z) > d(x, y) - \varepsilon = r.
  7. So z∉Bˉ(x,r)z \notin \bar{B}(x, r), that is, z∈Cz \in C. Thus B(y,ε)⊆CB(y, \varepsilon) \subseteq C.
  8. Each point of CC has a ball about it inside CC. By the description of the open sets of a metric space, CC is open.
  9. By the definition of a closed set, Bˉ(x,r)\bar{B}(x, r) is closed in XX.

Idea

A point outside the closed ball is farther than rr from the centre, by some margin. The triangle inequality shows that a ball of that margin about the point stays outside. So the complement is open.

Reading path

To understand this page from the ground up, read these 34 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 spacedefinitionClosed ball, definition, no evidence levelClosed balldefinitionOpen ball, definition, no evidence levelOpen balldefinitionTriangle inequality for a metric, statement, no evidence levelTriangle inequality f…statementMetric 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 balls are closed, proof, no evidence levelProof that closed bal…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. Closed ball definition
  27. Open ball definition
  28. Triangle inequality for a metric statement
  29. Metric topology definition
  30. Open set definition
  31. Proof of the description of the open sets of a metric space proof
  32. Closed set definition
  33. Open sets of a metric space statement
  34. Proof that closed balls are closed proof

Where this leads

62 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Closed_Ball_is_Closed_in_Metric_Space
About this page
Tag3XT4TV88MK
URIm2://3XT4TV88MK
Kindproof
WritingW0 stub
MSC54E35
ProvesClosed balls are closed
UsesMetric space, Closed ball, Open ball, Triangle inequality for a metric, Open sets of a metric space, Closed set
Changed4 November 2026

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