Math 2.0

General topology Definition

Open ball

Builds on Metric space

Definition

Let (X,d)(X, d) be a metric space, let x∈Xx \in X and let r>0r > 0 be a real number. The open ball with centre xx and radius rr is the set B(x,r)={y∈X:d(x,y)<r}.B(x, r) = \{ y \in X : d(x, y) < r \}.

Remarks

The centre is in the ball: x∈B(x,r)x \in B(x, r), because d(x,x)=0<rd(x, x) = 0 < r. In ℝ\mathbb{R} with the usual metric, B(x,r)B(x, r) is the open interval (x−r,x+r)(x - r, x + r). In the discrete metric, B(x,r)={x}B(x, r) = \{x\} for r≤1r \le 1 and B(x,r)=XB(x, r) = X for r>1r > 1. Some authors also allow r≤0r \le 0. Then B(x,r)B(x, r) is empty.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinition
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  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Implication definition
  6. Predicate definition
  7. Ordered pair definition
  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
  12. Uniqueness quantifier definition
  13. Binary relation definition
  14. Function definition
  15. Metric space definition
  16. Open ball definition

Where this leads

268 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ball_(mathematics), In general metric spaces
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Kinddefinition
WritingW1 stated
MSC54E35
DefinesOpen ball
UsesMetric space

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