Math 2.0

General topology Proposition

Composition of continuous maps is continuous

Builds on Continuous map of topological spacesFunction composition

Statement

Let XX, YY and ZZ be topological spaces. Let f ⁣:X→Yf \colon X \to Y and g ⁣:Y→Zg \colon Y \to Z be continuous maps. Then the composition g∘f ⁣:X→Zg \circ f \colon X \to Z is continuous.

Remarks

Proof: for each open set WW of ZZ, (g∘f)−1(W)=f−1(g−1(W))(g \circ f)^{-1}(W) = f^{-1}(g^{-1}(W)). The set g−1(W)g^{-1}(W) is open in YY, so its preimage under ff is open in XX. For example, x↦sin⁡(x2)x \mapsto \sin(x^2) is continuous on ℝ\mathbb{R}, because x↦x2x \mapsto x^2 and sin⁡\sin are 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 spacedefinitionFunction, definition, no evidence levelFunctiondefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionPreimage 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 setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionComposition of continuous maps is continuous, statement, no evidence levelComposition of contin…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.
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  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. Function definition
  23. Function composition definition
  24. Preimage of a set definition
  25. Metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Continuous map of topological spaces definition
  30. Composition of continuous maps is continuous statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Properties
About this page
Tag05NK26G0DR
URIm2://05NK26G0DR
Kindstatement, proposition
WritingW0 stub
MSC54C05
StatusThis copy of the site has no build data. Run m2 status 05NK26G0DR.
UsesContinuous map of topological spaces, Function composition

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