Math 2.0

General topology Definition

Totally bounded set

Builds on Metric spaceDiameter of a set

Definition

Let (X,d)(X, d) be a metric space and let S⊆XS \subseteq X. The set SS is totally bounded when for each real ε>0\varepsilon > 0 there are finitely many sets A1,…,An⊆XA_1, \ldots, A_n \subseteq X with S⊆A1∪⋯∪AnS \subseteq A_1 \cup \cdots \cup A_n and diam⁡(Ak)<ε\operatorname{diam}(A_k) < \varepsilon for each kk. The space XX is totally bounded when XX itself is a totally bounded set.

Remarks

Here diam⁡\operatorname{diam} is the diameter. Each totally bounded set is bounded. The converse is false: an infinite set with the discrete metric is bounded but not totally bounded. In ℝn\mathbb{R}^n with the Euclidean metric, a set is totally bounded if and only if it is bounded. The epsilon-net characterisation gives an equivalent form with balls.

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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 setdefinitionTotally bounded set, definition, no evidence levelTotally bounded setdefinition
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  1. Element definition
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  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
  20. Totally bounded set definition

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17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Totally_bounded_space, In metric spaces
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WritingW1 stated
MSC54E35
DefinesTotally bounded set
UsesMetric space, Diameter of a set

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