Math 2.0

General topology Theorem

Continuity via open sets

Builds on Continuous map of topological spacesOpen sets of a metric spaceEpsilon-delta continuity in metric spaces

Statement

Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces and let f ⁣:X→Yf \colon X \to Y be a function. Then the following two conditions are equivalent:

  1. For each a∈Xa \in X and each ε>0\varepsilon > 0 there is δ>0\delta > 0 such that dX(x,a)<δd_X(x, a) < \delta implies dY(f(x),f(a))<εd_Y(f(x), f(a)) < \varepsilon for all x∈Xx \in X.
  2. For each open set VV of YY, the preimage f−1(V)f^{-1}(V) is open in XX.

Remarks

Condition 2 is the definition of a continuous map between topological spaces. Thus, for metric spaces, the epsilon-delta definition and the topological definition agree. Proof that 1 implies 2: let a∈f−1(V)a \in f^{-1}(V). Choose ε\varepsilon with B(f(a),ε)⊆VB(f(a), \varepsilon) \subseteq V, then δ\delta from condition 1. Then B(a,δ)⊆f−1(V)B(a, \delta) \subseteq f^{-1}(V). Images of open sets need not be open: the constant map ℝ→ℝ\mathbb{R} \to \mathbb{R}, x↦0x \mapsto 0, is continuous, but the image of ℝ\mathbb{R} is {0}\{0\}.

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 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 setdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionOpen sets characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…statementContinuity at a point of a topological space, definition, no evidence levelContinuity at a point…definitionEpsilon-delta continuity in metric spaces, statement, no evidence levelEpsilon-delta continu…statementContinuity via open sets, statement, no evidence levelContinuity via open s…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. 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. Open sets of a metric space statement
  29. Continuous map of topological spaces definition
  30. Open sets characterised by neighbourhoods statement
  31. Continuity at a point of a topological space definition
  32. Epsilon-delta continuity in metric spaces statement
  33. Continuity via open sets statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Continuous functions between metric spaces
  2. Web https://en.wikipedia.org/wiki/Continuous_function, Continuous functions between topological spaces
About this page
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Kindstatement, theorem
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0YF82GBTND.
UsesContinuous map of topological spaces, Open sets of a metric space, Epsilon-delta continuity in metric spaces

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