Math 2.0

General topology Definition

First-countable space

Builds on Topological spaceOpen sets characterised by neighbourhoodsCountable set

Definition

Let XX be a topological space. The space XX is first-countable when each point x∈Xx \in X has a sequence (Nn)n≥1(N_n)_{n \ge 1} of neighbourhoods with this property: each neighbourhood of xx contains some NnN_n.

Remarks

Such a sequence is a countable neighbourhood basis at xx. One can choose the sets NnN_n open, and also decreasing. Each metric space is first-countable: take Nn=B(x,1/n)N_n = B(x, 1/n), as in open balls form a basis. Each second-countable space is first-countable. In a first-countable space, a point is in the closure of a set AA exactly when it is a limit of a sequence in AA.

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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 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 setdefinitionOpen sets characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…statementFirst-countable space, definition, no evidence levelFirst-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. 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. Injective function definition
  24. Metric space definition
  25. Countable set definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Open sets characterised by neighbourhoods statement
  30. First-countable space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/First-countable_space, lead section
About this page
Tag0HG11FW8MJ
URIm2://0HG11FW8MJ
Kinddefinition
WritingW1 stated
MSC54D70
DefinesFirst-countable space
UsesTopological space, Open sets characterised by neighbourhoods, Countable set

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