Math 2.0

General topology Definition

Subspace metric

Builds on Metric spaceSubsetRestriction of a function

Definition

Let (X,d)(X, d) be a metric space and let AA be a subset of XX. The subspace metric on AA is the function dA ⁣:A×A→ℝd_A \colon A \times A \to \mathbb{R} with dA(a,b)=d(a,b)d_A(a, b) = d(a, b) for all a,b∈Aa, b \in A. It is the restriction of dd to A×AA \times A. The metric space (A,dA)(A, d_A) is a subspace of (X,d)(X, d).

Remarks

The function dAd_A is a metric, because each axiom for dAd_A is a special case of the same axiom for dd. For example, the interval [0,1][0, 1] with d(x,y)=∣x−y∣d(x, y) = |x - y| is a subspace of the real line. The open sets of (A,dA)(A, d_A) are the sets U∩AU \cap A with UU open in XX. Thus the topology of dAd_A is the subspace topology.

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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 levelFunctiondefinitionRestriction of a function, definition, no evidence levelRestriction of a func…definitionMetric space, definition, no evidence levelMetric spacedefinitionSubspace metric, definition, no evidence levelSubspace metricdefinition
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  15. Restriction of a function definition
  16. Metric space definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Metric_space, Subspaces
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DefinesSubspace metric
UsesMetric space, Subset, Restriction of a function

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