Math 2.0

Field theory and polynomials Definition

Topological field

Builds on FieldTopological spaceContinuous function

Definition

Let KK be a field with a topology. Give K×KK \times K the product topology and K×=K∖{0}K^\times = K \setminus \set{0} the subspace topology. The field KK is a topological field if the following maps are continuous:

  1. addition K×K→KK \times K \to K, (x,y)↦x+y(x, y) \mapsto x + y;
  2. negation K→KK \to K, x↦−xx \mapsto -x;
  3. multiplication K×K→KK \times K \to K, (x,y)↦xy(x, y) \mapsto xy;
  4. inversion K×→K×K^\times \to K^\times, x↦x−1x \mapsto x^{-1}.

Remarks

Clauses 1 to 3 say that KK is a topological ring. Many authors also require the topology to be Hausdorff and not discrete. Example: ℝ\R with the usual topology is a topological field. Clauses 1 to 3 are the limit rules for sums, negatives and products, and clause 4 holds because x↦1/xx \mapsto 1/x is continuous on ℝ∖{0}\R \setminus \set{0}. More generally, every field with an absolute value, with the topology of the metric d(x,y)=∣x−y∣d(x, y) = \abs{x - y}, is a topological field. Non-example: ℝ\R with the lower limit topology, generated by the intervals [a,b)[a, b), is not a topological field. Negation fails clause 2: the preimage of the open set [0,1)[0, 1) under x↦−xx \mapsto -x is (−1,0](-1, 0], which is not open. Every field is a topological field for the discrete topology and for the indiscrete topology.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionGroup, concept, no evidence levelGroupconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionRing, definition, no evidence levelRingdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionConjunction, 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 setsdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionUnion, definition, no evidence levelUniondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionField, definition, no evidence levelFielddefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionContinuous function, definition, no evidence levelContinuous functiondefinitionTopological space, definition, no evidence levelTopological spacedefinitionTopological field, definition, no evidence levelTopological fielddefinition
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. Limit point of a set definition
  4. Proposition concept
  5. Set definition
  6. Ring definition
  7. Limit of a function definition
  8. Conjunction definition
  9. Disjunction definition
  10. Implication definition
  11. Predicate definition
  12. Empty set object
  13. Family of sets definition
  14. Commutative ring definition
  15. Intersection definition
  16. Unit of a ring definition
  17. Union definition
  18. Continuity at a point definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Field definition
  22. Subset definition
  23. Union of a family of sets definition
  24. Continuous function definition
  25. Topological space definition
  26. Topological field definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Field_(mathematics)
About this page
TagH4FS63VY0Y
URIm2://H4FS63VY0Y
Kinddefinition
WritingW0 stub
MSC12J99, 54H13
DefinesTopological field
UsesField, Topological space, Continuous function
Changed2 November 2026

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