Math 2.0

General topology Theorem

Continuous real function on a compact set attains its bounds

Builds on Continuous image of a compact set is compactHeine–Borel theoremSupremum

Statement

Let XX be a topological space, let K⊆XK \subseteq X be a nonempty compact set and let f ⁣:X→ℝf \colon X \to \mathbb{R} be a continuous map. Then there are points a,b∈Ka, b \in K with f(a)≤f(x)≤f(b)f(a) \le f(x) \le f(b) for all x∈Kx \in K.

Remarks

Proof: the set f(K)f(K) is compact in ℝ\mathbb{R}, because the continuous image of a compact set is compact. By the Heine–Borel theorem, it is closed and bounded. Thus it contains its supremum and its infimum. For K=[a,b]K = [a, b], this is the extreme value theorem of calculus. Compactness is needed: f(x)=xf(x) = x on (0,1)(0, 1) has no maximum, and f(x)=xf(x) = x on ℝ\mathbb{R} is not bounded.

Reading path

To understand this page from the ground up, read these 44 pages in order. The foundations come first.

  1. Element definition
  2. Euclidean metric example
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Supremum definition
  12. Empty set object
  13. Family of sets definition
  14. Ordered pair definition
  15. Intersection definition
  16. Union definition
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. Union of a family of sets definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Universal set definition
  25. Topological space definition
  26. Complement definition
  27. Function definition
  28. Subspace topology definition
  29. Preimage of a set definition
  30. Metric space definition
  31. Closed ball definition
  32. Open ball definition
  33. Bounded set definition
  34. Metric topology definition
  35. Open set definition
  36. Closed set definition
  37. Continuous map of topological spaces definition
  38. Open cover definition
  39. Compact space definition
  40. Compact set definition
  41. Heine–Borel theorem statement
  42. Compactness via open covers statement
  43. Continuous image of a compact set is compact statement
  44. Continuous real function on a compact set attains its bounds statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Extreme_value_theorem, Generalization to metric and topological spaces
About this page
Tag0V4DET59NW
URIm2://0V4DET59NW
Aliasesextreme-value-theorem
Kindstatement, theorem
WritingW0 stub
MSC54C30
StatusThis copy of the site has no build data. Run m2 status 0V4DET59NW.
UsesContinuous image of a compact set is compact, Heine–Borel theorem, Supremum

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