Math 2.0

General topology Definition

Distance from a point to a set

Builds on Metric spaceInfimum

Definition

Let (X,d)(X, d) be a metric space, let AA be a nonempty subset of XX and let x∈Xx \in X. The distance from xx to AA is the real number d(x,A)=inf⁡{d(x,a):a∈A}.d(x, A) = \inf \{ d(x, a) : a \in A \}.

Remarks

The infimum exists, because the set is nonempty and bounded below by 00. If x∈Ax \in A, then d(x,A)=0d(x, A) = 0. The converse is false: in ℝ\mathbb{R}, d(0,(0,1))=0d(0, (0, 1)) = 0 but 0∉(0,1)0 \notin (0, 1). In general d(x,A)=0d(x, A) = 0 if and only if xx is in the closure of AA. For all x,y∈Xx, y \in X, ∣d(x,A)−d(y,A)∣≤d(x,y)|d(x, A) - d(y, A)| \le d(x, y). Thus x↦d(x,A)x \mapsto d(x, A) is Lipschitz with constant 11.

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Element, definition, no evidence levelElementdefinitionProposition, 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 levelPredicatedefinitionInfimum, definition, no evidence levelInfimumdefinitionOrdered 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 spacedefinitionDistance from a point to a set, definition, no evidence levelDistance from a point…definition
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  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set bounded above definition
  5. Set definition
  6. Implication definition
  7. Predicate definition
  8. Infimum definition
  9. Ordered pair definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Uniqueness quantifier definition
  15. Binary relation definition
  16. Function definition
  17. Metric space definition
  18. Distance from a point to a set definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hausdorff_distance, Definition
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MSC54E35
DefinesDistance from a point to a set
UsesMetric space, Infimum

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