Math 2.0

General topology Separation axioms Lemma

Urysohn's lemma

Builds on Normal spaceContinuous map of topological spacesClosed set

Statement

Let XX be a normal space and let A,B⊆XA, B \subseteq X be disjoint closed sets. Then there is a continuous map f ⁣:X→[0,1]f \colon X \to [0, 1] with f(a)=0f(a) = 0 for all a∈Aa \in A and f(b)=1f(b) = 1 for all b∈Bb \in B.

Remarks

The converse also holds: if such a function exists for all disjoint closed AA and BB, then XX is normal. Take U=f−1([0,1/2))U = f^{-1}([0, 1/2)) and V=f−1((1/2,1])V = f^{-1}((1/2, 1]). The proof builds open sets UrU_r for the dyadic rationals r∈[0,1]r \in [0, 1], nested by the order of rr. Then f(x)f(x) is the infimum of the rr with x∈Urx \in U_r. In a metric space, the function f(x)=d(x,A)/(d(x,A)+d(x,B))f(x) = d(x, A) / (d(x, A) + d(x, B)) works. The Tietze extension theorem extends this result to all continuous functions on closed sets.

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To understand this page from the ground up, read these 33 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 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 spacedefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionClosed set, definition, no evidence levelClosed setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionNormal space, definition, no evidence levelNormal spacedefinitionUrysohn's lemma, statement, no evidence levelUrysohn's lemmastatement
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. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Metric space definition
  26. Preimage of a set definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Closed set definition
  31. Continuous map of topological spaces definition
  32. Normal space definition
  33. Urysohn's lemma statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Urysohn%27s_lemma, Formal statement
About this page
Tag01EY3VR7ZD
URIm2://01EY3VR7ZD
Aliasesurysohn-lemma
Kindstatement, lemma
WritingW1 stated
MSC54D15
StatusThis copy of the site has no build data. Run m2 status 01EY3VR7ZD.
UsesNormal space, Continuous map of topological spaces, Closed set
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