Math 2.0

General topology Proposition

Metrizable spaces are normal

Builds on Normal spaceMetrizable spaceDistance from a point to a setMetric spaces are Hausdorff

Statement

Let XX be a metrizable space. Then XX is normal.

Remarks

Proof: choose a metric dd for the topology. For disjoint closed sets AA and BB, let U={x:d(x,A)<d(x,B)}U = \{x : d(x, A) < d(x, B)\} and V={x:d(x,B)<d(x,A)}V = \{x : d(x, B) < d(x, A)\}. Here d(x,A)d(x, A) is the distance to a set. The sets UU and VV are open, because x↦d(x,A)x \mapsto d(x, A) is continuous. They are disjoint, and A⊆UA \subseteq U because d(x,A)=0<d(x,B)d(x, A) = 0 < d(x, B) for x∈Ax \in A. Single points are closed in a metric space. Thus each metrizable space is normal and T1, and also Hausdorff.

Reading path

To understand this page from the ground up, read these 38 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 bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionInfimum, definition, no evidence levelInfimumdefinitionEmpty 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 relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, 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…statementDistance from a point to a set, definition, no evidence levelDistance from a point…definitionMetric topology, definition, no evidence levelMetric topologydefinitionMetrizable space, definition, no evidence levelMetrizable spacedefinitionOpen set, definition, no evidence levelOpen setdefinitionClosed set, definition, no evidence levelClosed setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionMetric spaces are Hausdorff, statement, no evidence levelMetric spaces are Hau…statementNormal space, definition, no evidence levelNormal spacedefinitionMetrizable spaces are normal, statement, no evidence levelMetrizable spaces are…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 bounded above definition
  5. Set definition
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Predicate definition
  10. Infimum definition
  11. Empty set object
  12. Family of sets definition
  13. Ordered pair definition
  14. Intersection definition
  15. Union definition
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Subset definition
  20. Union of a family of sets definition
  21. Uniqueness quantifier definition
  22. Binary relation definition
  23. Universal set definition
  24. Topological space definition
  25. Complement definition
  26. Function definition
  27. Metric space definition
  28. Open ball definition
  29. Triangle inequality for a metric statement
  30. Distance from a point to a set definition
  31. Metric topology definition
  32. Metrizable space definition
  33. Open set definition
  34. Closed set definition
  35. Hausdorff space definition
  36. Metric spaces are Hausdorff statement
  37. Normal space definition
  38. Metrizable spaces are normal statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_space, Examples of normal spaces
About this page
Tag0AD0BZ49DW
URIm2://0AD0BZ49DW
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0AD0BZ49DW.
UsesNormal space, Metrizable space, Distance from a point to a set, Metric spaces are Hausdorff

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