Math 2.0

General topology Proof

Proof of the intermediate value theorem for connected spaces

Builds on Connected spaceConnected setSubspace topologyContinuous map of topological spacesContinuous image of a connected set is connectedConnected subsets of the real line are intervals

Proof

We combine the two facts that images of connected sets are connected and that connected subsets of ℝ\mathbb{R} are intervals.

  1. The subspace topology on XX as a subset of itself is the topology of XX. So XX is a connected subset of XX, because XX is a connected space.
  2. The map ff is continuous. By continuous image of a connected set is connected, the set f(X)⊆ℝf(X) \subseteq \mathbb{R} is connected.
  3. By connected subsets of the real line are intervals, f(X)f(X) is an interval. So for p,q∈f(X)p, q \in f(X) and p<c<qp < c < q, we have c∈f(X)c \in f(X).
  4. Let cc be between f(a)f(a) and f(b)f(b). If c=f(a)c = f(a), take x=ax = a. If c=f(b)c = f(b), take x=bx = b.
  5. Otherwise cc lies strictly between the two values. Let pp be the smaller and qq the larger of f(a)f(a) and f(b)f(b). Then p,q∈f(X)p, q \in f(X) and p<c<qp < c < q.
  6. By step 3, c∈f(X)c \in f(X). So there is x∈Xx \in X with f(x)=cf(x) = c.

Idea

The image of a connected space under a continuous map is connected. In ℝ\mathbb{R} the connected sets are the intervals, and an interval that contains f(a)f(a) and f(b)f(b) contains each value between them.

Reading path

To understand this page from the ground up, read these 64 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof that the continuous image of a connected set is connected, Proof of the description of the open sets of a metric space, Proof that connected subsets of the real line are intervals.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Rational numbers object
  7. Real numbers object
  8. Set bounded above definition
  9. Set definition
  10. Ring of integers statement
  11. Conjunction definition
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. The reals form a field statement
  16. Supremum definition
  17. Empty set object
  18. Family of sets definition
  19. Ordered pair definition
  20. Intersection definition
  21. Union definition
  22. The rationals form a field statement
  23. Existential quantifier definition
  24. Universal quantifier definition
  25. Completeness axiom statement
  26. Cartesian product definition
  27. Subset definition
  28. Union of a family of sets definition
  29. The rationals form an ordered field statement
  30. Uniqueness quantifier definition
  31. Binary relation definition
  32. Universal set definition
  33. Order on the reals definition
  34. Topological space definition
  35. Antisymmetric relation definition
  36. Reflexive relation definition
  37. Transitive relation definition
  38. Complement definition
  39. Function definition
  40. Interval definition
  41. Subspace topology definition
  42. Preorder definition
  43. Preimage of a set definition
  44. Metric space definition
  45. Partial order definition
  46. Open ball definition
  47. Total order definition
  48. Metric topology definition
  49. Open set definition
  50. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  51. Proof of the description of the open sets of a metric space proof
  52. Continuous map of topological spaces definition
  53. Closed set definition
  54. The reals form an ordered field statement
  55. Open sets of a metric space statement
  56. Connected space definition
  57. Proof of the approximation property of the supremum proof
  58. Connected set definition
  59. Approximation property of the supremum statement
  60. Proof that the continuous image of a connected set is connected proof
  61. Proof that connected subsets of the real line are intervals proof
  62. Continuous image of a connected set is connected statement
  63. Connected subsets of the real line are intervals statement
  64. Proof of the intermediate value theorem for connected spaces proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Intermediate_value_theorem, General metric and topological spaces
About this page
Tag6G8JWR1YN6
URIm2://6G8JWR1YN6
Kindproof
WritingW0 stub
MSC54D05
ProvesIntermediate value theorem for connected spaces
UsesConnected space, Connected set, Subspace topology, Continuous map of topological spaces, Continuous image of a connected set is connected, Connected subsets of the real line are intervals
Changed4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.