>>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.