Math 2.0

General topology Proposition

Projections are continuous

Builds on Product topologyContinuous map of topological spacesOpen map

Statement

Let XX and YY be topological spaces and give X×YX \times Y the product topology. Then the projections π1 ⁣:X×Y→X\pi_1 \colon X \times Y \to X, (x,y)↦x(x, y) \mapsto x, and π2 ⁣:X×Y→Y\pi_2 \colon X \times Y \to Y, (x,y)↦y(x, y) \mapsto y, are continuous.

Remarks

Proof: for UU open in XX, we have π1−1(U)=U×Y\pi_1^{-1}(U) = U \times Y, which is a basic open set. The proof for π2\pi_2 is the same. The projections are also open maps, but in general they are not closed maps. The product topology is the coarsest topology on X×YX \times Y that makes both projections continuous.

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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 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 topologydefinitionImage of a set, definition, no evidence levelImage of a setdefinitionMetric space, definition, no evidence levelMetric spacedefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen map, definition, no evidence levelOpen mapdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionProjections are continuous, statement, no evidence levelProjections are conti…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. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Basis of a topology definition
  23. Function definition
  24. Product topology definition
  25. Image of a set definition
  26. Metric space definition
  27. Preimage of a set definition
  28. Open ball definition
  29. Metric topology definition
  30. Open set definition
  31. Open map definition
  32. Continuous map of topological spaces definition
  33. Projections are continuous statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Product_topology, Properties
About this page
Tag0PM88Z9W4M
URIm2://0PM88Z9W4M
Kindstatement, proposition
WritingW0 stub
MSC54B10
StatusThis copy of the site has no build data. Run m2 status 0PM88Z9W4M.
UsesProduct topology, Continuous map of topological spaces, Open map

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