Math 2.0

General topology Proposition

Continuity via closed sets

Builds on Continuous map of topological spacesClosed setComplement of an open set is closedPreimage of a set

Statement

Let XX and YY be topological spaces and let f ⁣:X→Yf \colon X \to Y be a function. Then ff is continuous if and only if for each closed set FF of YY, the preimage f−1(F)f^{-1}(F) is closed in XX.

Remarks

Proof: preimages commute with complements, that is, f−1(Y∖A)=X∖f−1(A)f^{-1}(Y \setminus A) = X \setminus f^{-1}(A) for each A⊆YA \subseteq Y. Then use that the complements of open sets are closed. For example, for a continuous f ⁣:ℝ→ℝf \colon \mathbb{R} \to \mathbb{R}, the set of zeros f−1({0})f^{-1}(\{0\}) is closed.

Reading path

To understand this page from the ground up, read these 33 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 relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric 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 setdefinitionClosed set, definition, no evidence levelClosed setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionComplement of an open set is closed, statement, no evidence levelComplement of an open…statementContinuity via closed sets, statement, no evidence levelContinuity via closed…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. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Metric space definition
  26. Preimage of a set definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Closed set definition
  31. Continuous map of topological spaces definition
  32. Complement of an open set is closed statement
  33. Continuity via closed sets statement

Where this leads

22 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Closure operator and interior operator definitions
About this page
Tag0B8GPXFSP7
URIm2://0B8GPXFSP7
Kindstatement, proposition
WritingW0 stub
MSC54C05
StatusThis copy of the site has no build data. Run m2 status 0B8GPXFSP7.
UsesContinuous map of topological spaces, Closed set, Complement of an open set is closed, Preimage of a set

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