|
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: the statement can neither be proven nor disproven from those axioms. (Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.) ==Formulation== Given a non-empty totally ordered set ''R'' with the following four properties: # ''R'' does not have a least nor a greatest element; # the order on ''R'' is dense (between any two 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; # 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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Suslin's problem」の詳細全文を読む スポンサード リンク
|