Suslin's hypothesis
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin (1920) and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.
(Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.)
Un ensemble ordonné (linéairement) sans sauts ni lacunes et tel que tout ensemble de ses intervalles (contenant plus qu'un élément) n'empiétant pas les uns sur les autres est au plus dénumerable, est-il nécessairement un continue linéaire (ordinaire)?
A (linearly) ordered set without jumps or gaps and such that every set of its intervals (containing more than one element) not overlapping each other is at most denumerable, is it necessarily an (ordinary) linear continuum?
Given a non-empty totally ordered set R with the following four properties:
Is R necessarily order-isomorphic to the real line R?
If the requirement for the countable chain condition is replaced with the requirement that R contains a countable dense subset (i.e., R is a separable space) then the answer is indeed yes: any such set R is necessarily order-isomorphic to R (proved by Cantor).
The condition for a topological space that every collection of non-empty disjoint open sets is at most countable is called the Suslin property.
Any totally ordered set that is not isomorphic to R but satisfies (1) – (4) is known as a Suslin line. The Suslin hypothesis says that there are no Suslin lines: that every countable-chain-condition dense complete linear order without endpoints is isomorphic to the real line. An equivalent statement is that every tree of height ω1 either has a branch of length ω1 or an antichain of cardinality
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