Math 2.0

General topology Definition

Tychonoff space

Builds on Topological spaceT1 spaceClosed setContinuous functionRegular spaceNormal spaceUrysohn's lemma

Definition

Let XX be a topological space. XX is completely regular when the following holds. For every closed set F⊆XF \subseteq X and every point x∈X∖Fx \in X \setminus F, there is a continuous function f ⁣:X→[0,1]f \colon X \to [0, 1] with f(x)=0f(x) = 0 and f(y)=1f(y) = 1 for all y∈Fy \in F. XX is a Tychonoff space when it is completely regular and T1.

Remarks

In the definition, T1T_1 can be replaced by T0T_0, because a completely regular T0T_0 space is T1T_1. A Tychonoff space is regular and Hausdorff: the sets f−1([0,1/2))f^{-1}([0, 1/2)) and f−1((1/2,1])f^{-1}((1/2, 1]) separate xx from FF. A space is Tychonoff if and only if it is homeomorphic to a subspace of a product [0,1]I[0, 1]^I. By Urysohn's lemma, every normal T1T_1 space is Tychonoff.

Example: every metric space (X,d)(X, d) is Tychonoff. For x∉Fx \notin F with FF closed, d(x,F)>0d(x, F) > 0, and f(y)=min⁡(1,d(y,x)/d(x,F))f(y) = \min\bigl(1, d(y, x) / d(x, F)\bigr) is continuous, equal to 00 at xx and to 11 on FF.

Non-example: an infinite set XX with the cofinite topology is T1T_1 but not completely regular. Each continuous f ⁣:X→ℝf \colon X \to \R is constant: two distinct values would have disjoint open preimages, but two nonempty open sets of XX always meet. So no ff separates a point xx from the closed set {y}\set{y} with y≠xy \ne x.

Notation: T312T_{3\frac12}.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

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Closed set, concept, no evidence levelClosed setconceptContinuous function, concept, no evidence levelContinuous functionconceptElement, definition, no evidence levelElementdefinitionNormal space, concept, no evidence levelNormal spaceconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectRegular space, concept, no evidence levelRegular spaceconceptT1 space, concept, no evidence levelT1 spaceconceptTopological space, concept, no evidence levelTopological spaceconceptSet, 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 levelFunctiondefinitionPreimage 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 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 lemmastatementTychonoff space, definition, no evidence levelTychonoff spacedefinition
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. Closed set concept
  2. Continuous function concept
  3. Element definition
  4. Normal space concept
  5. Proposition concept
  6. Real numbers object
  7. Regular space concept
  8. T1 space concept
  9. Topological space concept
  10. Set definition
  11. Conjunction definition
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Empty set object
  16. Family of sets definition
  17. Ordered pair definition
  18. Intersection definition
  19. Union definition
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. Union of a family of sets definition
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Topological space definition
  29. Complement definition
  30. Function definition
  31. Preimage of a set definition
  32. Metric space definition
  33. Open ball definition
  34. Metric topology definition
  35. Open set definition
  36. Closed set definition
  37. Continuous map of topological spaces definition
  38. Normal space definition
  39. Urysohn's lemma statement
  40. Tychonoff space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Tychonoff_space
About this page
Tag76M4XEGPAB
URIm2://76M4XEGPAB
Kinddefinition
WritingW0 stub
MSC54D15
DefinesTychonoff space
UsesTopological space, T1 space, Closed set, Continuous function, Regular space, Normal space, Urysohn's lemma
Changed2 November 2026

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