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Concept
Suslin's problem
Summary
In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously.
It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; 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.)
right
Suslin's problem asks: Given a non-empty totally ordered set R with the four properties
R does not have a least nor a greatest element;
the order on R is dense (between any two distinct elements there is another);
the order on R is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and
every collection of mutually disjoint non-empty open intervals in R is countable (this is the countable chain condition for the order topology of R),
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 properties 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 .
The generalized Suslin hypothesis says that for every infinite regular cardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ. The existence of Suslin lines is equivalent to the existence of Suslin trees and to Suslin algebras.
Official source
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