Math 2.0

General topology Definition

Second-countable space

Builds on Topological spaceBasis of a topologyCountable set

Definition

A topological space (X,τ)(X, \tau) is second-countable when τ\tau has a basis ℬ\mathcal{B} that is countable.

Remarks

The space ℝn\mathbb{R}^n is second-countable: the open balls with rational centres and rational radii form a countable basis. Each second-countable space is separable and first-countable. A metric space is second-countable if and only if it is separable. An uncountable set with the discrete topology is not second-countable.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, 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 spacedefinitionBasis of a topology, definition, no evidence levelBasis of a topologydefinitionFunction, definition, no evidence levelFunctiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionCountable set, definition, no evidence levelCountable setdefinitionSecond-countable space, definition, no evidence levelSecond-countable 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. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Family of sets definition
  10. Ordered pair definition
  11. Intersection definition
  12. Union definition
  13. Existential quantifier definition
  14. Universal quantifier definition
  15. Cartesian product definition
  16. Subset definition
  17. Union of a family of sets definition
  18. Uniqueness quantifier definition
  19. Binary relation definition
  20. Topological space definition
  21. Basis of a topology definition
  22. Function definition
  23. Injective function definition
  24. Countable set definition
  25. Second-countable space definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Second-countable_space, lead section
About this page
Tag00RE1R4W7N
URIm2://00RE1R4W7N
Kinddefinition
WritingW1 stated
MSC54D70
DefinesSecond-countable space
UsesTopological space, Basis of a topology, Countable set

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