{"title":"度量度量空间和极大\\(\\delta \\)分离集中的选择公理","authors":"Michał Dybowski, Przemysław Górka","doi":"10.1007/s00153-023-00868-4","DOIUrl":null,"url":null,"abstract":"<div><p>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 Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal <span>\\(\\delta \\)</span>-separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00868-4.pdf","citationCount":"0","resultStr":"{\"title\":\"The axiom of choice in metric measure spaces and maximal \\\\(\\\\delta \\\\)-separated sets\",\"authors\":\"Michał Dybowski, Przemysław Górka\",\"doi\":\"10.1007/s00153-023-00868-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>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 Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal <span>\\\\(\\\\delta \\\\)</span>-separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.</p></div>\",\"PeriodicalId\":48853,\"journal\":{\"name\":\"Archive for Mathematical Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2023-03-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00153-023-00868-4.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Archive for Mathematical Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00153-023-00868-4\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Arts and Humanities\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-023-00868-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 0
摘要
我们证明了可数选择公理是证明伪度量空间上Borel测度的存在性的必要和充分的，使得开球的测度是正的和有限的，这意味着该空间的可分性。通过这种方式，给出了用Górka（Am Math Mon 128:84-2020）公式化的一个开放问题的否定答案。此外，我们还从选择公理及其弱形式的角度研究了度量空间和伪度量空间中极大分离集的存在性。
The axiom of choice in metric measure spaces and maximal \(\delta \)-separated sets
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 Górka (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.
期刊介绍：
The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.
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