Math 2.0

General topology Definition

Quotient topology

Builds on Topological spaceEquivalence relationQuotient setPreimage of a set

Definition

Let (X,τ)(X, \tau) be a topological space and let ∼\sim be an equivalence relation on XX. Let q ⁣:X→X/∼q \colon X \to X / {\sim} be the map to the quotient set that sends xx to its class. The quotient topology on X/∼X / {\sim} is {V⊆X/∼:q−1(V)∈τ}.\{ V \subseteq X / {\sim} : q^{-1}(V) \in \tau \}.

Remarks

It is the finest topology on X/∼X / {\sim} for which qq is continuous. A preimage q−1(V)q^{-1}(V) is a union of classes. Thus VV is open if and only if the union of the classes in VV is open in XX. Quotients need not keep good properties: a quotient of a Hausdorff space need not be Hausdorff. Examples are the circle, the torus and the Möbius strip.

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To understand this page from the ground up, read these 30 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, 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 relationdefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionEquivalence class, definition, no evidence levelEquivalence classdefinitionQuotient set, definition, no evidence levelQuotient setdefinitionQuotient topology, definition, no evidence levelQuotient topologydefinition
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. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Family of sets definition
  10. Ordered pair definition
  11. Intersection definition
  12. Union definition
  13. Set-builder notation concept
  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. Reflexive relation definition
  23. Symmetric relation definition
  24. Transitive relation definition
  25. Function definition
  26. Equivalence relation definition
  27. Preimage of a set definition
  28. Equivalence class definition
  29. Quotient set definition
  30. Quotient topology definition

Where this leads

23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_space_(topology), Definition
About this page
Tag0V8046R0P9
URIm2://0V8046R0P9
Kinddefinition
WritingW1 stated
MSC54B15
DefinesQuotient topology
UsesTopological space, Equivalence relation, Quotient set, Preimage of a set

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