Math 2.0

General topology Proposition

Metric spaces are Hausdorff

Builds on Hausdorff spaceOpen ballTriangle inequality for a metricMetric topology

Statement

Let (X,d)(X, d) be a metric space, with the metric topology. Then XX is a Hausdorff space.

Remarks

Proof: for x≠yx \neq y, let r=d(x,y)/2>0r = d(x, y) / 2 > 0. The open balls B(x,r)B(x, r) and B(y,r)B(y, r) are open. They are disjoint: a point zz in both would give d(x,y)≤d(x,z)+d(z,y)<2rd(x, y) \le d(x, z) + d(z, y) < 2r, by the triangle inequality. Thus each metrizable space is Hausdorff. A non-Hausdorff space, such as an infinite set with the cofinite topology, is not metrizable. For a pseudometric, the statement fails: points at distance 00 cannot be separated.

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To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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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 balldefinitionTriangle inequality for a metric, statement, no evidence levelTriangle inequality f…statementMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionMetric spaces are Hausdorff, statement, no evidence levelMetric spaces are Hau…statement
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. Hausdorff space definition
  29. Metric spaces are Hausdorff statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hausdorff_space, Examples of Hausdorff and non-Hausdorff spaces
About this page
Tag0P9DCDES73
URIm2://0P9DCDES73
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0P9DCDES73.
UsesHausdorff space, Open ball, Triangle inequality for a metric, Metric topology

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