Math 2.0

General topology Separation axioms Theorem

Tietze extension theorem

Builds on Normal spaceUrysohn's lemmaContinuous map of topological spacesRestriction of a function

Statement

Let XX be a normal space, let A⊆XA \subseteq X be closed, and let f ⁣:A→ℝf \colon A \to \mathbb{R} be continuous for the subspace topology on AA. Then there is a continuous F ⁣:X→ℝF \colon X \to \mathbb{R} whose restriction to AA is ff. If also ∣f(a)∣≤M|f(a)| \le M for all a∈Aa \in A, then one can choose FF with ∣F(x)∣≤M|F(x)| \le M for all x∈Xx \in X.

Remarks

The proof applies Urysohn's lemma many times and sums a uniformly convergent series of functions. The converse holds: if each continuous function from each closed subset to ℝ\mathbb{R} extends, then XX is normal. The closedness of AA is needed: the function sin⁡(1/x)\sin(1/x) on (0,1](0, 1] has no continuous extension to [0,1][0, 1].

Reading path

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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 setdefinitionRestriction of a function, definition, no evidence levelRestriction of a func…definitionOpen 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 lemmastatementTietze extension theorem, statement, no evidence levelTietze extension theo…statement
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  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. Restriction of a function definition
  28. Open ball definition
  29. Metric topology definition
  30. Open set definition
  31. Closed set definition
  32. Continuous map of topological spaces definition
  33. Normal space definition
  34. Urysohn's lemma statement
  35. Tietze extension theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Tietze_extension_theorem, Formal statement
About this page
Tag0X52TNMTCJ
URIm2://0X52TNMTCJ
Aliasestietze-urysohn-extension-theorem
Kindstatement, theorem
WritingW1 stated
MSC54C20
StatusThis copy of the site has no build data. Run m2 status 0X52TNMTCJ.
UsesNormal space, Urysohn's lemma, Continuous map of topological spaces, Restriction of a function
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