Math 2.0

General topology Theorem

Universal property of the quotient topology

Builds on Quotient topologyContinuous map of topological spacesFunction composition

Statement

Let XX be a topological space, let ∼\sim be an equivalence relation on XX, and give X/∼X / {\sim} the quotient topology, with quotient map q ⁣:X→X/∼q \colon X \to X / {\sim}. Let YY be a topological space and let g ⁣:X/∼→Yg \colon X / {\sim} \to Y be a function. Then gg is continuous if and only if the composition g∘q ⁣:X→Yg \circ q \colon X \to Y is continuous.

Remarks

Proof of the converse: for VV open in YY, the set q−1(g−1(V))=(g∘q)−1(V)q^{-1}(g^{-1}(V)) = (g \circ q)^{-1}(V) is open in XX. Thus g−1(V)g^{-1}(V) is open in the quotient. So each continuous map f ⁣:X→Yf \colon X \to Y that is constant on each class gives a unique continuous map gg with f=g∘qf = g \circ q. For example, t↦(cos⁡2πt,sin⁡2πt)t \mapsto (\cos 2\pi t, \sin 2\pi t) is constant on the classes of ℝ/ℤ\mathbb{R} / \mathbb{Z}, so it gives a continuous map from ℝ/ℤ\mathbb{R} / \mathbb{Z} to the circle.

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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 levelUniondefinitionSet-builder notation, concept, no evidence levelSet-builder notationconceptExistential 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 spacedefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionSymmetric relation, definition, no evidence levelSymmetric relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionEquivalence relation, definition, no evidence levelEquivalence relationdefinitionMetric space, definition, no evidence levelMetric spacedefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionOpen ball, definition, no evidence levelOpen balldefinitionEquivalence class, definition, no evidence levelEquivalence classdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionQuotient set, definition, no evidence levelQuotient setdefinitionOpen set, definition, no evidence levelOpen setdefinitionQuotient topology, definition, no evidence levelQuotient topologydefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionUniversal property of the quotient topology, statement, no evidence levelUniversal property of…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. Set-builder notation concept
  15. Existential quantifier definition
  16. Universal quantifier definition
  17. Cartesian product definition
  18. Subset definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Topological space definition
  23. Reflexive relation definition
  24. Symmetric relation definition
  25. Transitive relation definition
  26. Function definition
  27. Equivalence relation definition
  28. Metric space definition
  29. Preimage of a set definition
  30. Function composition definition
  31. Open ball definition
  32. Equivalence class definition
  33. Metric topology definition
  34. Quotient set definition
  35. Open set definition
  36. Quotient topology definition
  37. Continuous map of topological spaces definition
  38. Universal property of the quotient topology statement

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_space_(topology), Properties
About this page
Tag0FF2FJ8CH2
URIm2://0FF2FJ8CH2
Kindstatement, theorem
WritingW0 stub
MSC54B15
StatusThis copy of the site has no build data. Run m2 status 0FF2FJ8CH2.
UsesQuotient topology, Continuous map of topological spaces, Function composition

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