Math 2.0

General topology Definition

Closed ball

Builds on Metric space

Definition

Let (X,d)(X, d) be a metric space, let x∈Xx \in X and let r≥0r \ge 0 be a real number. The closed ball with centre xx and radius rr is the set Bˉ(x,r)={y∈X:d(x,y)≤r}.\bar{B}(x, r) = \{ y \in X : d(x, y) \le r \}.

Remarks

The open ball B(x,r)B(x, r) is a subset of Bˉ(x,r)\bar{B}(x, r). In ℝ\mathbb{R} with the usual metric, Bˉ(x,r)\bar{B}(x, r) is the closed interval [x−r,x+r][x - r, x + r]. The closed ball is not always the closure of the open ball. In the discrete metric, B(x,1)={x}B(x, 1) = \{x\} is closed, but Bˉ(x,1)=X\bar{B}(x, 1) = X. For r=0r = 0, the closed ball is {x}\{x\}.

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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 spacedefinitionClosed ball, definition, no evidence levelClosed balldefinition
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Further reading

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

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