Math 2.0

General topology Proposition

Limits are unique in Hausdorff spaces

Builds on Hausdorff spaceOpen sets characterised by neighbourhoods

Statement

Let XX be a Hausdorff space and let (xn)n≥1(x_n)_{n \ge 1} be a sequence in XX. Say that (xn)(x_n) converges to x∈Xx \in X when for each neighbourhood NN of xx there is n0n_0 with xn∈Nx_n \in N for all n≥n0n \ge n_0. If (xn)(x_n) converges to xx and to yy, then x=yx = y.

Remarks

Proof: if x≠yx \neq y, take disjoint open sets U∋xU \ni x and V∋yV \ni y. For large nn, the point xnx_n is in both UU and VV, which is impossible. The same holds for nets and filters. In fact, a space is Hausdorff if and only if each net in it has at most one limit. In an indiscrete space with at least two points, each sequence converges to each point.

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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 setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionOpen sets characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…statementLimits are unique in Hausdorff spaces, statement, no evidence levelLimits are unique in …statement
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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. Hausdorff space definition
  28. Open sets characterised by neighbourhoods statement
  29. Limits are unique in Hausdorff spaces statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hausdorff_space, Equivalences
About this page
Tag0D8T984XEE
URIm2://0D8T984XEE
Kindstatement, proposition
WritingW0 stub
MSC54D10
StatusThis copy of the site has no build data. Run m2 status 0D8T984XEE.
UsesHausdorff space, Open sets characterised by neighbourhoods

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