Math 2.0

General topology Proof

Proof that the projections of a product are continuous

Builds on Product topologyBasis of a topologyContinuous map of topological spacesPreimage of a setTopological space

Proof

We compute preimages of open sets.

  1. Let UU be open in XX. The preimage is π1−1(U)={(x,y)∈X×Y:x∈U}=U×Y\pi_1^{-1}(U) = \set{(x, y) \in X \times Y : x \in U} = U \times Y.
  2. The set YY is open in YY by the axioms of a topology. So U×YU \times Y is one of the sets U×VU \times V of the product topology basis.
  3. By the definition of a basis, its members are open. So π1−1(U)\pi_1^{-1}(U) is open in X×YX \times Y, and π1\pi_1 is continuous.
  4. Let VV be open in YY. In the same way, π2−1(V)=X×V\pi_2^{-1}(V) = X \times V, and XX is open in XX.
  5. So X×VX \times V is a basic open set, and π2\pi_2 is continuous.

Idea

The preimage of an open set under a projection is a box whose other side is the whole factor. Such a box is one of the basic open sets of the product topology.

Reading path

To understand this page from the ground up, read these 31 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 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 topologydefinitionMetric 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 setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionProof that the projections of a product are continuous, proof, no evidence levelProof that the projec…proof
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. Metric space definition
  26. Preimage of a set definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Continuous map of topological spaces definition
  31. Proof that the projections of a product are continuous proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Product_topology, Definition
About this page
Tag45HZCDR9N7
URIm2://45HZCDR9N7
Kindproof
WritingW0 stub
MSC54B10
ProvesProjections are continuous
UsesProduct topology, Basis of a topology, Continuous map of topological spaces, Preimage of a set, Topological space
Changed4 November 2026

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