(参考) 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.
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.
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. The Axiom of Dependent Choices implies the Baire Category Theorem for complete pseudometric spaces, and the latter implies the Axiom of Countable Choice.
