The axiom of choice in metric measure spaces and maximal d -separated sets

被引：0
|
作者：
Dybowski, Michal [1 ]
Gorka, Przemyslaw [1 ]
机构：
[1] Warsaw Univ Technol, Dept Math & Informat Sci, Pl Politech 1, PL-00661 Warsaw, Poland
来源：
ARCHIVE FOR MATHEMATICAL LOGIC | 2023年 / 62卷 / 5-6期
关键词：
Axiom of choice; Dependent choice; Countable choice; delta-separated sets; Borel measure; Doubling measure; Doubling metric space;
D O I：
10.1007/s00153-023-00868-4
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in G & oacute;rka (Am Math Mon 128:84-86, 2020) is given. Moreover, we study existence of maximal delta -separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.
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页码：735 / 749
页数：15
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