Math 2.0

General topology Proof

Proof that metric spaces are Hausdorff

Builds on Hausdorff spaceOpen ballOpen balls are openTriangle inequality for a metricMetric space

Proof

We separate two points by balls of half their distance.

  1. Let x,y∈Xx, y \in X with x≠yx \neq y. By the axioms of a metric, d(x,y)>0d(x, y) > 0. Let r=d(x,y)/2>0r = d(x, y) / 2 > 0.
  2. The open balls B(x,r)B(x, r) and B(y,r)B(y, r) are open. They contain xx and yy, since d(x,x)=0<rd(x, x) = 0 < r and d(y,y)=0<rd(y, y) = 0 < r.
  3. Suppose some zz lies in both balls. Then d(x,z)<rd(x, z) < r and d(z,y)<rd(z, y) < r.
  4. By the triangle inequality, d(x,y)≤d(x,z)+d(z,y)<2r=d(x,y)d(x, y) \le d(x, z) + d(z, y) < 2r = d(x, y). This is impossible.
  5. So B(x,r)∩B(y,r)=∅B(x, r) \cap B(y, r) = \varnothing, and XX is Hausdorff.

Idea

Two distinct points are at a positive distance. Balls of half that radius around them cannot meet, since a common point would give a path between them that is shorter than their distance.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first. The path takes the proofs Proof of the description of the open sets of a metric space, Proof that open balls are open.

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 balldefinitionTriangle inequality for a metric, statement, no evidence levelTriangle inequality f…statementMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionProof of the description of the open sets of a metric space, proof, no evidence levelProof of the descript…proofHausdorff space, definition, no evidence levelHausdorff spacedefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementProof that open balls are open, proof, no evidence levelProof that open balls…proofOpen balls are open, statement, no evidence levelOpen balls are openstatementProof that metric spaces are Hausdorff, proof, no evidence levelProof that metric spa…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. Triangle inequality for a metric statement
  26. Metric topology definition
  27. Open set definition
  28. Proof of the description of the open sets of a metric space proof
  29. Hausdorff space definition
  30. Open sets of a metric space statement
  31. Proof that open balls are open proof
  32. Open balls are open statement
  33. Proof that metric spaces are Hausdorff proof

Where this leads

704 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Metric_Space_is_Hausdorff
About this page
TagNK3WS9ENHG
URIm2://NK3WS9ENHG
Kindproof
WritingW0 stub
MSC54E35
ProvesMetric spaces are Hausdorff
UsesHausdorff space, Open ball, Open balls are open, Triangle inequality for a metric, Metric space
Changed4 November 2026

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