[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ13 (1002ﾚｽ)
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409: 現代数学の系譜 雑談 ◆yH25M02vWFhP 02/09(日)09:54 ID:lz6oAIdr(4/12) AAS
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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