Math 2.0

Associative rings and algebras Definition

Topological ring

Builds on RingTopological spaceContinuous map of topological spacesProduct topology

Definition

Let RR be a ring and let 𝒯\mathcal{T} be a topology on the set RR. Give RΓ—RR \times R the product topology. The pair (R,𝒯)(R, \mathcal{T}) is a topological ring if the following maps are continuous:

  1. addition RΓ—Rβ†’RR \times R \to R, (a,b)↦a+b(a, b) \mapsto a + b;
  2. negation Rβ†’RR \to R, aβ†¦βˆ’aa \mapsto -a;
  3. multiplication RΓ—Rβ†’RR \times R \to R, (a,b)↦ab(a, b) \mapsto ab.

Remarks

Clauses 1 and 2 say that (R,+)(R, +) is a topological group. Example: ℝ\R with the usual topology; continuity of (a,b)↦ab(a, b) \mapsto ab follows from abβˆ’aβ€²bβ€²=a(bβˆ’bβ€²)+(aβˆ’aβ€²)bβ€²ab - a'b' = a(b - b') + (a - a')b'. Example: any ring with the discrete topology, since every map from a discrete space is continuous. Example: for an ideal II of a commutative ring RR, the II-adic topology, with the ideals InI^n as a base of neighbourhoods of 00. Non-example: ℝ\R with the lower limit topology, whose base is the intervals [a,b)[a, b). Negation is not continuous: the preimage of the open set [0,1)[0, 1) is (βˆ’1,0](-1, 0], which is not open there.

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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 levelElementdefinitionGroup, concept, no evidence levelGroupconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionRing, definition, no evidence levelRingdefinitionConjunction, 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 relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionBasis of a topology, definition, no evidence levelBasis of a topologydefinitionFunction, definition, no evidence levelFunctiondefinitionProduct topology, definition, no evidence levelProduct topologydefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionTopological ring, definition, no evidence levelTopological ringdefinition
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. Group concept
  3. Proposition concept
  4. Real numbers object
  5. Set definition
  6. Ring definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate 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. Topological space definition
  24. Basis of a topology definition
  25. Function definition
  26. Product topology definition
  27. Preimage of a set definition
  28. Metric space definition
  29. Open ball definition
  30. Metric topology definition
  31. Open set definition
  32. Continuous map of topological spaces definition
  33. Topological ring definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Topological_ring
About this page
TagCXV1969XGB
URIm2://CXV1969XGB
Kinddefinition
WritingW0 stub
MSC16W80, 54H13
DefinesTopological ring
UsesRing, Topological space, Continuous map of topological spaces, Product topology
Changed2 November 2026

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