Math 2.0

Mathematical logic and foundations Object

Suslin line

Builds on Total orderDense orderSeparable spaceSuslin's problem

Description

A Suslin line is a nonempty totally ordered set (L,<)(L, <) with these properties.

  1. LL has no least element and no greatest element.
  2. The order is dense.
  3. The order is complete: each nonempty subset of LL that is bounded above has a least upper bound.
  4. Each family of pairwise disjoint nonempty open intervals of LL is countable.
  5. LL is not separable: no countable subset of LL meets every nonempty open interval.

Remarks

The real line satisfies clauses 1 to 4 but not clause 5, so ℝ\R is not a Suslin line. In fact, a total order with clauses 1 to 3 and separability is order isomorphic to ℝ\R. So a Suslin line is exactly a counterexample to Suslin's problem. The existence of a Suslin line is independent of ZFC, if ZFC is consistent. The diamond principle implies that a Suslin line exists. Martin's axiom together with the failure of the continuum hypothesis implies that no Suslin line exists. A Suslin line exists if and only if a Suslin tree exists.

Reading path

To understand this page from the ground up, read these 47 pages in order. The foundations come first.

  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Supremum concept
  5. Total order concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Linear continuum 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. Intersection of a family of sets definition
  22. Union of a family of sets definition
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Topological space definition
  27. Antisymmetric relation definition
  28. Reflexive relation definition
  29. Transitive relation definition
  30. Complement definition
  31. Function definition
  32. Preorder definition
  33. Metric space definition
  34. Injective function definition
  35. Partial order definition
  36. Open ball definition
  37. Countable set definition
  38. Dense order definition
  39. Total order definition
  40. Metric topology definition
  41. Open set definition
  42. Closed set definition
  43. Closure definition
  44. Dense set definition
  45. Separable space definition
  46. Suslin's problem problem
  47. Suslin line object

Further reading

  1. Web https://en.wikipedia.org/wiki/Suslin%27s_problem
About this page
TagMEKFR4Q617
URIm2://MEKFR4Q617
Kindobject
WritingW0 stub
MSC03E50, 06A05, 03E35
Instance ofTotal order
UsesTotal order, Dense order, Separable space, Suslin's problem
Changed2 November 2026

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