Math 2.0

General topology Definition

Continuous map of topological spaces

Builds on Topological spaceOpen setPreimage of a set

Definition

Let (X,τX)(X, \tau_X) and (Y,τY)(Y, \tau_Y) be topological spaces and let f ⁣:X→Yf \colon X \to Y be a function. The function ff is continuous when for each open set V∈τYV \in \tau_Y, the preimage f−1(V)f^{-1}(V) is in τX\tau_X.

Remarks

Equivalently, ff is continuous if and only if preimages of closed sets are closed, as in continuity via closed sets. For metric spaces, this is the same as the epsilon-delta condition, by continuity via open sets. A function is continuous if and only if it is continuous at each point. Constant maps and identity maps are continuous, and the composition of continuous maps is continuous.

Reading path

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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 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 setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definition
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. Preimage of a set definition
  24. Metric space definition
  25. Open ball definition
  26. Metric topology definition
  27. Open set definition
  28. Continuous map of topological spaces definition

Where this leads

86 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Continuous functions between topological spaces
About this page
Tag0SY9ECQ19Q
URIm2://0SY9ECQ19Q
Aliasescontinuous-map
Kinddefinition
WritingW0 stub
MSC54C05
DefinesContinuous map
UsesTopological space, Open set, Preimage of a set

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