Math 2.0

General topology Connectedness Theorem

Connected subsets of the real line are intervals

Builds on Connected setIntervalSupremumCompleteness axiom

Statement

Let S⊆ℝS \subseteq \mathbb{R}, where ℝ\mathbb{R} has the usual topology. Then SS is connected if and only if SS is an interval, that is, for all a,b∈Sa, b \in S and all c∈ℝc \in \mathbb{R} with a<c<ba < c < b, c∈Sc \in S.

Remarks

If SS is not an interval, take a<c<ba < c < b with a,b∈Sa, b \in S and c∉Sc \notin S. Then (−∞,c)(-\infty, c) and (c,∞)(c, \infty) split SS. The proof that an interval II is connected uses the completeness axiom. If open sets UU and VV split II, with a∈Ua \in U, b∈Vb \in V and a<ba < b, let cc be the supremum of the points of U∩[a,b]U \cap [a, b]. Then cc lies in neither UU nor VV. Under this definition the empty set and single points are intervals, and they are connected. The statement fails for ℚ\mathbb{Q}: no interval of ℚ\mathbb{Q} with more than one point is connected.

Reading path

To understand this page from the ground up, read these 45 pages in order. The foundations come first.

  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. Supremum definition
  16. Empty set object
  17. Family of sets definition
  18. Ordered pair definition
  19. Intersection definition
  20. Union definition
  21. The rationals form a field statement
  22. Existential quantifier definition
  23. Universal quantifier definition
  24. Completeness axiom statement
  25. Cartesian product definition
  26. Subset definition
  27. Union of a family of sets definition
  28. The rationals form an ordered field statement
  29. Uniqueness quantifier definition
  30. Binary relation definition
  31. Universal set definition
  32. Order on the reals definition
  33. Topological space definition
  34. Complement definition
  35. Function definition
  36. Interval definition
  37. Subspace topology definition
  38. Metric space definition
  39. Open ball definition
  40. Metric topology definition
  41. Open set definition
  42. Closed set definition
  43. Connected space definition
  44. Connected set definition
  45. Connected subsets of the real line are intervals statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Connected_space, Examples
About this page
Tag0B9ZQFMQNV
URIm2://0B9ZQFMQNV
Kindstatement, theorem
WritingW0 stub
MSC54D05
StatusThis copy of the site has no build data. Run m2 status 0B9ZQFMQNV.
UsesConnected set, Interval, Supremum, Completeness axiom

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

Used by