Math 2.0

General topology Definition

Diameter of a set

Builds on Metric spaceSupremumExtended real number line

Definition

Let (X,d)(X, d) be a metric space and let SS be a nonempty subset of XX. The diameter of SS is diam⁡(S)=sup⁡{d(x,y):x,y∈S},\operatorname{diam}(S) = \sup \{ d(x, y) : x, y \in S \}, where the supremum is taken in the extended real number line, so that diam⁡(S)∈[0,∞]\operatorname{diam}(S) \in [0, \infty].

Remarks

The set SS is bounded if and only if diam⁡(S)<∞\operatorname{diam}(S) < \infty. For the empty set, a common convention is diam⁡(∅)=0\operatorname{diam}(\emptyset) = 0. In ℝ\mathbb{R}, the interval (a,b)(a, b) with a<ba < b has diameter b−ab - a, and so does [a,b][a, b]. If S⊆TS \subseteq T, then diam⁡(S)≤diam⁡(T)\operatorname{diam}(S) \le \operatorname{diam}(T).

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Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionOrdered 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 spacedefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinition
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  1. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Supremum definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Metric space definition
  19. Diameter of a set definition

Where this leads

94 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Diameter, Generalizations
About this page
Tag0692CGQF00
URIm2://0692CGQF00
Kinddefinition
WritingW0 stub
MSC54E35
DefinesDiameter of a set
UsesMetric space, Supremum, Extended real number line

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