Math 2.0

General topology Proposition

Closed balls are closed

Builds on Closed ballClosed setOpen sets of a metric spaceTriangle inequality for a metric

Statement

Let (X,d)(X, d) be a metric space, let x∈Xx \in X and let r≥0r \ge 0. Then the closed ball Bˉ(x,r)\bar{B}(x, r) is a closed set of XX.

Remarks

Proof: let y∉Bˉ(x,r)y \notin \bar{B}(x, r) and let ε=d(x,y)−r>0\varepsilon = d(x, y) - r > 0. If d(y,z)<εd(y, z) < \varepsilon, then d(x,z)≥d(x,y)−d(y,z)>rd(x, z) \ge d(x, y) - d(y, z) > r by the triangle inequality. Thus the complement of Bˉ(x,r)\bar{B}(x, r) is open. For r=0r = 0, this shows that each one-point set {x}\{x\} is closed in a metric space.

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To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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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 setdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementClosed set, definition, no evidence levelClosed setdefinitionClosed balls are closed, statement, no evidence levelClosed balls are clos…statement
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. Open sets of a metric space statement
  32. Closed set definition
  33. Closed balls are closed statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Ball_(mathematics), In general metric spaces
About this page
Tag0YS40B38JD
URIm2://0YS40B38JD
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0YS40B38JD.
UsesClosed ball, Closed set, Open sets of a metric space, Triangle inequality for a metric

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