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ガロア第一論文と乗数イデアル他関連資料スレ13

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>>409
> つづき
> 
> There exist models of ZF that violate the above conditions ([17], [18]).
> Observe the fine distinction between conditions 2 and 3 of Theorem 1.1.
> These may lead one to assume that also the following property is equivalent to the above conditions:
> (*) a function f : R −→ R is continuous iff it is sequentially continuous.
> However, this would be a serious mistake: (*) holds in ZF (without any choiceassumptions) — see [29].
> If, however, we consider functions f : X −→ R with metric domain we need even more choice than in Theorem 1.1, — see Theorem 2.1.
> Proposition 1.2 ([15]). Equivalent are:
> 1. in R, every bounded infinite set contains a convergent injective sequence,
> 2. every infinite subset of R is Dedekind-infinite.
> There exist models of ZF that violate the above conditions ([18]).
> Obviously, the conditions of Theorem 1.1 imply the conditions of Proposition 1.2.
> Is the converse true?
> Observe that the following slight modifications of condition 1 in Proposition 1.2 hold in ZF:
> (a) in R, every bounded countable set contains a convergent injective sequence,
> (b) in R, for every bounded infinite set there exists an accumulation point.
> 
> ＜Lindelöfとは？＞
> en.wikipedia.org/wiki/Lindel%C3%B6f_space
> Lindelöf space
> In mathematics, a Lindelöf space[1][2] is a topological space in which every open cover has a countable subcover.
> The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.
> 
> （注：上記の”(*) a function f : R −→ R is continuous iff it is sequentially continuous. (*) holds in ZF (without any choiceassumptions) — see [29]”が、下記と思う）
> alg-d.com/math/ac/continuous.html
> トップ > 数学 > 選択公理 > 実数関数の連続性
> 壱大整域 20130323
> 一方，次の命題はZFで証明できる．
> 命題 f: R→Rとする．
> fがRで連続 ⇔ 収束点列 { xn }n=0∞に対して limn→∞f(xn) = f(limn→∞xn)
> 証明 略す
> 
> ja.wikipedia.org/wiki/%E5%AE%9F%E6%95%B0%E3%81%AE%E9%80%A3%E7%B6%9A%E6%80%A7
> 実数の連続性（continuity of real numbers）とは、実数の集合がもつ性質である。有理数はこの性質を持たない。
> 実数の連続性は、実数の完備性 (completeness of the real numbers) とも言われる
> (引用終り)
> 以上

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