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Compact Metric Spaces and Weak Forms of the Axiom of Choice
Author(s) -
Keremedis Kyriakos,
Tachtsis Eleftherios
Publication year - 2001
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/1521-3870(200101)47:1<117::aid-malq117>3.0.co;2-n
Subject(s) - mathematics , axiom of choice , countable set , zermelo–fraenkel set theory , separable space , metric space , compact space , statement (logic) , axiom , metric (unit) , convex metric space , nowhere dense set , injective metric space , pure mathematics , discrete mathematics , set (abstract data type) , set theory , mathematical analysis , geometry , operations management , computer science , economics , programming language , political science , law
It is shown that for compact metric spaces ( X, d ) the following statements are pairwise equivalent: “ X is Loeb”, “ X is separable”, “ X has a we ordered dense subset”, “ X is second countable”, and “ X has a dense set G = ∪{ G n : n ∈ ω }, ∣ G n ∣ < ω , with lim n →∞ diam ( G n ) = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF 0 , the Zermelo‐Fraenkel set theory without the axiom of regularity, and that the countable axiom of choice for families of finite sets CAC fin does not imply the statement “Compact metric spaces are separable”.