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.

引用

页码：735 / 749

页数：15

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