[過去ﾛｸﾞ] 現代数学の系譜 カントル 超限集合論他 3 (548ﾚｽ)
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39(3): 2020/07/27(月)14:54 ID:dppBRBhf(3/5) AAS
>>37
補足

Frechet filterの英wikipedia記事と
”Examples
On the set N of natural numbers, the set of infinite intervals B = { (n,∞) : n ∈ N} is a Frechet filter base, i.e., the Frechet filter on N consists of all supersets of elements of B.”
あと、MathWorld
”Cofinite Filter
If S is an infinite set, then the collection F_S={ A ⊆ S:S-A is finite} is a filter called the cofinite (or Frechet) filter on S.”

（参考）
外部ﾘﾝｸ:en.wikipedia.org
省17

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40(1): 2020/07/27(月)15:07 ID:dppBRBhf(4/5) AAS
>>39
＞ supersets

補足
supersetは、subsetの逆だな
あまり使わないかも

外部ﾘﾝｸ:ejje.weblio.jp
Weblio記号和英辞書での「superset of」の意味
superset of
記号：⊃
(真)部分集合の逆方向
省6

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42(4): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/07/27(月)21:41 ID:slbIBvLt(1/6) AAS
>>39 補足
外部ﾘﾝｸ[pdf]:arxiv.org
Filters and Ultrafilters in Real Analysis 2012
Max Garcia Mathematics Department California Polytechnic State University

Abstract
We study free filters and their maximal extensions on the set of natural numbers.
We characterize the limit of a sequence of real numbers in terms of the Fr´echet filter, which involves only one quantifier as opposed to the three non-commuting quantifiers in the usual definition.
We construct the field of real non-standard numbers and study their properties.
We characterize the limit of a sequence of real numbers in terms of non-standard numbers which only requires a single quantifier as well.
We are trying to make the point that the involvement of filters and/or non-standard numbers leads to a reduction in the number of quantifiers and hence, simplification, compared to the more traditional ε, δ-definition of limits in real analysis.
省17

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49(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/07/27(月)23:39 ID:slbIBvLt(5/6) AAS
>>39
補足

外部ﾘﾝｸ:en.wikipedia.org
Frechet filter
より

”Examples
On the set N of natural numbers, the set of infinite intervals B = { (n,∞) : n ∈ N} is a Frechet filter base, i.e., the Frechet filter on N consists of all supersets of elements of B.[citation needed]”

このExampleは、時枝無関係でしょ
つまり、
”On the set N of natural numbers, the set of infinite intervals B = { (n,∞) : n ∈ N} is a Frechet filter base,”
省5

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