Math 2.0

General topology Proposition

Open sets of a metric space

Builds on Metric spaceOpen ballOpen setMetric topology

Statement

Let (X,d)(X, d) be a metric space and let U⊆XU \subseteq X. Then UU is open in the metric topology if and only if for each x∈Ux \in U there is a real ε>0\varepsilon > 0 with B(x,ε)⊆UB(x, \varepsilon) \subseteq U. Here B(x,ε)B(x, \varepsilon) is the open ball with centre xx and radius ε\varepsilon.

Remarks

The number ε\varepsilon can depend on xx. In ℝ\mathbb{R}, the set (0,∞)(0, \infty) is open: for x>0x > 0, take ε=x\varepsilon = x. The set ℚ\mathbb{Q} is not open in ℝ\mathbb{R}, because each ball about a rational number contains an irrational number.

Reading path

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Metric space definition
  24. Open ball definition
  25. Metric topology definition
  26. Open set definition
  27. Open sets of a metric space statement

Where this leads

34 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Open_set, Metric space
About this page
Tag0A26X79RTB
URIm2://0A26X79RTB
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0A26X79RTB.
UsesMetric space, Open ball, Open set, Metric topology

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