Math 2.0

General topology Definition

Bounded set

Builds on Metric spaceClosed ball

Definition

Let (X,d)(X, d) be a metric space and let S⊆XS \subseteq X. The set SS is bounded when there is a real number MM such that d(x,y)≤Md(x, y) \le M for all x,y∈Sx, y \in S.

Remarks

The empty set and each finite set are bounded. If XX is not empty, then SS is bounded if and only if SS lies in some closed ball Bˉ(x,r)\bar{B}(x, r) with x∈Xx \in X and r≥0r \ge 0. In ℝ\mathbb{R} with the usual metric, SS is bounded in this sense if and only if SS is bounded above and bounded below. Boundedness depends on the metric: ℝ\mathbb{R} is not bounded for ∣x−y∣|x - y|, but it is bounded for min⁡(∣x−y∣,1)\min(|x - y|, 1).

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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 balldefinitionBounded set, definition, no evidence levelBounded setdefinition
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  1. Element definition
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  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
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  13. Binary relation definition
  14. Function definition
  15. Metric space definition
  16. Closed ball definition
  17. Bounded set definition

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8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bounded_set, Definition in a metric space
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WritingW1 stated
MSC54E35
DefinesBounded set
UsesMetric space, Closed ball

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