Math 2.0

General topology Proof

Proof that limits are unique in Hausdorff spaces

Builds on Hausdorff spaceOpen sets characterised by neighbourhoods

Proof

We argue by contradiction.

  1. Suppose (xn)(x_n) converges to xx and to yy, and x≠yx \neq y.
  2. Since XX is Hausdorff, there are open sets UU and VV with x∈Ux \in U, y∈Vy \in V and U∩V=∅U \cap V = \varnothing.
  3. The open set UU satisfies x∈U⊆Ux \in U \subseteq U, so UU is a neighbourhood of xx. In the same way, VV is a neighbourhood of yy.
  4. By convergence to xx, there is n0n_0 with xn∈Ux_n \in U for all n≥n0n \ge n_0. By convergence to yy, there is n1n_1 with xn∈Vx_n \in V for all n≥n1n \ge n_1.
  5. Let n=max⁡(n0,n1)n = \max(n_0, n_1). Then xn∈U∩Vx_n \in U \cap V, which contradicts U∩V=∅U \cap V = \varnothing.
  6. Hence x=yx = y.

Idea

Two different limits could be separated by disjoint open sets. A convergent sequence eventually stays in each of them, so from some index on it would lie in both at once, which is impossible.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first. The path takes the proof Proof that open sets are the sets that are neighbourhoods of all their points.

Show the path as a graph
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 spacedefinitionProof that open sets are the sets that are neighbourhoods of all their points, proof, no evidence levelProof that open sets …proofOpen sets characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…statementProof that limits are unique in Hausdorff spaces, proof, no evidence levelProof that limits are…proof
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. Metric space definition
  24. Open ball definition
  25. Metric topology definition
  26. Open set definition
  27. Hausdorff space definition
  28. Proof that open sets are the sets that are neighbourhoods of all their points proof
  29. Open sets characterised by neighbourhoods statement
  30. Proof that limits are unique in Hausdorff spaces proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hausdorff_space, Equivalences
About this page
TagS2F8B8CRGP
URIm2://S2F8B8CRGP
Kindproof
WritingW0 stub
MSC54D10
ProvesLimits are unique in Hausdorff spaces
UsesHausdorff space, Open sets characterised by neighbourhoods
Changed4 November 2026

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