ガロア第一論文と乗数イデアル他関連資料スレ12

[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12 (1002ﾚｽ)
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236: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/01/13(月) 18:15:11.17 ID:xSRlEtRO

つづき

Notes et références
3．Pour d'autres énoncés équivalents à ACω, voir (en) Horst Herrlich, « Choice principles in elementary topology and analysis », Comment. Math. Univ. Carolinae, vol. 38, no 3,‎ 1997, p. 545-552 (lire en ligne [archive]) et (en) Paul Howard et Jean E. Rubin, Consequences of the Axiom of Choice, Providence, R.I., AMS, 1998.

archive.wikiwix.com/cache/display2.php?url=http%3A%2F%2Fwww.emis.de%2Fjournals%2FCMUC%2Fpdf%2Fcmuc9703%2Fherrli.pdf
Comment.Math.Univ.Carolin. 38,3(1997)545–552 545
Choice principles in elementary topology and analysis Horst Herrlich
1. In the realm of the reals
We start by observing that several familiar topological properties of the reals are equivalent to each other and to rather natural choice-principles.
Theorem 1.1 ([15], [29], [30]). Equivalent are:
1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x,
2. a function f : R → R is continuous at a point x iff it is sequentially continuous at x,
3. a real-valued function f : A → R from a subspace A of R is continuous iff it is sequentially continuous,
4. each subspace of R is separable,
5. R is a Lindel¨ of space,
6. Q is a Lindel¨ of space,
7. N is a Lindel¨ of space,
8. each unbounded subset of R contains an unbounded sequence,
9. the Axiom of Choice for countable collections of subsets of R.
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.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1735693028/236

239: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/01/13(月) 19:08:04.14 ID:xSRlEtRO

>>235-236より

1）可算選択の公理なしで、コーシー列の収束が言えることと
　上記 fr.wikipedia 可算選択公理における下記の記述とは、矛盾しない と思う
”Theorem 1.1 ([15], [29], [30]). Equivalent are:
1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x,
2. a function f : R → R is continuous at a point x iff it is sequentially continuous at x,
3. a real-valued function f : A → R from a subspace A of R is continuous iff it is sequentially continuous,
4. each subspace of R is separable,
5. R is a Lindel¨ of space,
6. Q is a Lindel¨ of space,
7. N is a Lindel¨ of space,
8. each unbounded subset of R contains an unbounded sequence,
9. the Axiom of Choice for countable collections of subsets of R.
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.”

2）つまり、可算選択の公理なしで、コーシー列の収束が言えるとして
　その上で、可算選択公理を認めると
”1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x,”
”4. each subspace of R is separable,”
”5. R is a Lindel¨ of space,”
　成立！

3）というか、”9. the Axiom of Choice for countable collections of subsets of R.”
　と、Equivalent である！

つづく

http://rio2016.5ch.net/test/read.cgi/math/1735693028/239

270: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/01/14(火) 17:22:40.20 ID:rO5NkXOo

>>267
(引用開始)
>つまり、整列可能定理は公理として、有理コーシー列で有理数Qの完備化を可能として
>無理数（超越数を含む）の存在を保証する
は君の発言だよね？　食言ってことは、未だに間違いって理解してないってこと？
(引用終り)

では、下記の通り 微修正をします ；ｐ）

つまり、整列可能定理は公理として、有理コーシー列で有理数Qの完備化を可能として
　↓
つまり、整列可能定理は公理として、x∈R subset A⊂R で 有理コーシー列 a sequence in A＼{x} that converges to x で有理数Qの完備化を可能として(但し、RをcompactにするためDCを使用>>261)

(参考)
　>>236より下記（Equivalent are:1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x, ＆ 9. the Axiom of Choice for countable collections of subsets of R.）
archive.wikiwix.com/cache/display2.php?url=http%3A%2F%2Fwww.emis.de%2Fjournals%2FCMUC%2Fpdf%2Fcmuc9703%2Fherrli.pdf
Comment.Math.Univ.Carolin. 38,3(1997)545–552 545
Choice principles in elementary topology and analysis Horst Herrlich
1. In the realm of the reals
We start by observing that several familiar topological properties of the reals are equivalent to each other and to rather natural choice-principles.
Theorem 1.1 ([15], [29], [30]). Equivalent are:
1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x,
2. a function f : R → R is continuous at a point x iff it is sequentially continuous at x,
3. a real-valued function f : A → R from a subspace A of R is continuous iff it is sequentially continuous,
4. each subspace of R is separable,
5. R is a Lindel¨ of space,
6. Q is a Lindel¨ of space,
7. N is a Lindel¨ of space,
8. each unbounded subset of R contains an unbounded sequence,
9. the Axiom of Choice for countable collections of subsets of R.
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.

http://rio2016.5ch.net/test/read.cgi/math/1735693028/270

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