Math 2.0

General topology Theorem

Universal property of the product topology

Builds on Product topology on an arbitrary productContinuous map of topological spacesFunction compositionSubbasis

Statement

Let (Xi)i∈I(X_i)_{i \in I} be a family of topological spaces, let X=∏i∈IXiX = \prod_{i \in I} X_i have the product topology, and let πi ⁣:X→Xi\pi_i \colon X \to X_i be the projections. Let ZZ be a topological space and let f ⁣:Z→Xf \colon Z \to X be a function. Then ff is continuous if and only if the composition πi∘f ⁣:Z→Xi\pi_i \circ f \colon Z \to X_i is continuous for each i∈Ii \in I.

Remarks

Proof of the converse: preimages of the subbasic sets πi−1(U)\pi_i^{-1}(U) under ff are (πi∘f)−1(U)(\pi_i \circ f)^{-1}(U), which are open. Preimages respect unions and intersections, so all preimages of open sets are open. Thus a map into a product is continuous if and only if all its coordinate functions are continuous. For example, t↦(cos⁡t,sin⁡t)t \mapsto (\cos t, \sin t) is continuous from ℝ\mathbb{R} to ℝ2\mathbb{R}^2. The statement fails for the box topology.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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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 levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionIntersection of a family of sets, definition, no evidence levelIntersection of a fam…definitionUnion 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 levelFunctiondefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionSubbasis, definition, no evidence levelSubbasisdefinitionOpen ball, definition, no evidence levelOpen balldefinitionProduct topology on an arbitrary product, definition, no evidence levelProduct topology on a…definitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionUniversal property of the product topology, statement, no evidence levelUniversal property of…statement
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. 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. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Intersection of a family of sets definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Topological space definition
  23. Basis of a topology definition
  24. Function definition
  25. Function composition definition
  26. Metric space definition
  27. Preimage of a set definition
  28. Subbasis definition
  29. Open ball definition
  30. Product topology on an arbitrary product definition
  31. Metric topology definition
  32. Open set definition
  33. Continuous map of topological spaces definition
  34. Universal property of the product topology statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Product_topology, Properties
About this page
Tag0M6W4XNST6
URIm2://0M6W4XNST6
Kindstatement, theorem
WritingW0 stub
MSC54B10
StatusThis copy of the site has no build data. Run m2 status 0M6W4XNST6.
UsesProduct topology on an arbitrary product, Continuous map of topological spaces, Function composition, Subbasis

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