>>242 (引用開始) 3）とすると、(assuming countable choice) ならば、>>239より ”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,” だから、不足しているのは Rが ”Metric” であることだが。”Rが Metric”をいうには、countable choice だけでは 不足なのかな？ (引用終り)
下記 Construction of the real numbers の Construction from Cauchy sequences で metric spaces として completion（完備）までやっているが、どの選択公理を使うかの記述がない ”axiom of dependent choice”だと思うのだが・・ （＾＾
(参考) https://en.wikipedia.org/wiki/Construction_of_the_real_numbers Construction of the real numbers Explicit constructions of models
Construction from Cauchy sequences A standard procedure to force all Cauchy sequences in a metric space to converge is adding new points to the metric space in a process called completion. R is defined as the completion of the set Q of the rational numbers with respect to the metric |x − y| Normally, metrics are defined with real numbers as values, but this does not make the construction/definition circular, since all numbers that are implied (even implicitly) are rational numbers.[5]
An advantage of constructing R as the completion of Q is that this construction can be used for every other metric spaces.
