Math 2.0

General topology Definition

Separable space

Builds on Dense setCountable set

Definition

A topological space XX is separable when there is a countable subset D⊆XD \subseteq X that is dense in XX.

Remarks

The real line is separable, because ℚ\mathbb{Q} is countable and dense. In the same way ℝn\mathbb{R}^n is separable, with the dense set ℚn\mathbb{Q}^n. Each second-countable space is separable. For metric spaces, the converse is also true. An uncountable set with the discrete metric is not separable, because each dense subset is the whole set.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first.

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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 levelSubsetdefinitionIntersection of a family of sets, definition, no evidence levelIntersection of a fam…definitionUnion 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 relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionCountable set, definition, no evidence levelCountable setdefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionClosed set, definition, no evidence levelClosed setdefinitionClosure, definition, no evidence levelClosuredefinitionDense set, definition, no evidence levelDense setdefinitionSeparable space, definition, no evidence levelSeparable spacedefinition
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.
  • no evidence level
  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. Intersection of a family of sets definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Universal set definition
  23. Topological space definition
  24. Complement definition
  25. Function definition
  26. Injective function definition
  27. Metric space definition
  28. Countable set definition
  29. Open ball definition
  30. Metric topology definition
  31. Open set definition
  32. Closed set definition
  33. Closure definition
  34. Dense set definition
  35. Separable space definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Separable_space, lead section
About this page
Tag0GKJ78948V
URIm2://0GKJ78948V
Kinddefinition
WritingW1 stated
MSC54D65
DefinesSeparable space
UsesDense set, Countable set

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