Math 2.0

General topology Definition

Pseudometric space

Builds on Metric space

Definition

Let XX be a set. A pseudometric on XX is a function d ⁣:X×X→ℝd \colon X \times X \to \mathbb{R} such that, for all x,y,z∈Xx, y, z \in X:

  1. d(x,x)=0d(x, x) = 0.
  2. d(x,y)=d(y,x)d(x, y) = d(y, x).
  3. d(x,z)≤d(x,y)+d(y,z)d(x, z) \le d(x, y) + d(y, z).

A pseudometric space is a pair (X,d)(X, d), where XX is a set and dd is a pseudometric on XX.

Remarks

Each metric is a pseudometric. A pseudometric is a metric if and only if d(x,y)=0d(x, y) = 0 implies x=yx = y. On the set of functions f ⁣:[0,1]→ℝf \colon [0, 1] \to \mathbb{R}, the rule d(f,g)=∣f(0)−g(0)∣d(f, g) = |f(0) - g(0)| is a pseudometric, but not a metric. The relation d(x,y)=0d(x, y) = 0 is an equivalence relation, and dd gives a metric on the set of its classes.

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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 spacedefinitionPseudometric space, definition, no evidence levelPseudometric spacedefinition
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  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Implication definition
  6. Predicate definition
  7. Ordered pair definition
  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
  12. Uniqueness quantifier definition
  13. Binary relation definition
  14. Function definition
  15. Metric space definition
  16. Pseudometric space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Pseudometric_space, Definition
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MSC54E35
DefinesPseudometric space
UsesMetric space

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