赤ペン先生、入ります！ｗｗ ；ｐ） 「なんで整列定理が必要と思ったの？」については、下記のHorst Herrlichの ”Theorem 2.4 ([4], [14]). Equivalent are: 1. in a (pseudo)metric space X, a point x is an accumulation point of a subset A iff there exists a sequence in A\ {x} that converges to x, 略す 17. the Axiom of Countable Choice.” を、百回音読してね ；ｐ）
なお、下記のソロヴェイモデル 到達不能基数＋ ”ZF + DC を満たしで 実数集合が全てルベーグ可測で perfect set property を持ち、ベールの性質を持つものになっている” ここの部分は、到達不能基数が ZFCの外です だから、到達不能基数＋”ZF + DC と、”17. the Axiom of Countable Choice”は、直ちには矛盾していないことを付言しておきます ；ｐ） （本音は、良く分からないｗ）
(参考) 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 P546 2. In the realm of pseudometric spaces In this section we consider (pseudo)metric spaces and various compactness-notions for them.
Theorem 2.1 ([4], [15]). Equivalent are: 1. every separable pseudometric space is a Lindel¨ of space, 2. every pseudometric space with a countable base is a Lindel¨ of space, 3. the Axiom of Choice for countable collections of subsets of R.
Definition 2.2. A pseudometric space X is called 1. Heine-Borel-compact provided every open cover of X contains a finite one, 2. Weierstraß-compact provided for every infinite subset of X there exists an accumulation point, 3. Alexandroff-Urysohn-compact provided for every infinite subset of X there exists a complete accumulation point, 4. sequentially-compact provided every sequence in X has a convergent subsequence. Under the Axiom of Choice the above compactness concepts are equivalent. This is no longer the case in ZF.
