{"title":"On lower density operators","authors":"Gertruda Ivanova, Elżbieta Wagner-Bojakowska","doi":"10.1515/ms-2024-0014","DOIUrl":null,"url":null,"abstract":"The classical density topology is an extension of the natural topology on the real line, as the interior of arbitrary Lebesgue measurable set <jats:italic>A</jats:italic> is contained in the set of density points of <jats:italic>A</jats:italic>. Also each density point of <jats:italic>A</jats:italic> belongs to the closure of <jats:italic>A</jats:italic> for arbitrary measurable set <jats:italic>A</jats:italic>. In this paper, we concentrate on lower density operators for which the inclusions mentioned above are not fulfilled. In the first part, examples of such lower density operators generated by measure-preserving bijections are given. There are introduced three conditions to investigate lower density operators for which only the second inclusion holds. In the second part, the concept of operator <jats:italic>D</jats:italic> introduced by K. Kuratowski is applied to the characterization of such operators.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":"59 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Slovaca","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/ms-2024-0014","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The classical density topology is an extension of the natural topology on the real line, as the interior of arbitrary Lebesgue measurable set A is contained in the set of density points of A. Also each density point of A belongs to the closure of A for arbitrary measurable set A. In this paper, we concentrate on lower density operators for which the inclusions mentioned above are not fulfilled. In the first part, examples of such lower density operators generated by measure-preserving bijections are given. There are introduced three conditions to investigate lower density operators for which only the second inclusion holds. In the second part, the concept of operator D introduced by K. Kuratowski is applied to the characterization of such operators.
经典密度拓扑学是实线上自然拓扑学的扩展，因为任意 Lebesgue 可测集 A 的内部包含在 A 的密度点集合中。在第一部分中，我们举例说明了由保度量双射产生的此类低密度算子。本文引入了三个条件来研究只有第二个包含成立的低密度算子。在第二部分中，K. Kuratowski 引入的算子 D 概念被应用于此类算子的表征。
期刊介绍：
Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process. Its reputation was approved by many outstanding mathematicians who already contributed to Math. Slovaca. It makes bridges among mathematics, physics, soft computing, cryptography, biology, economy, measuring, etc. The Journal publishes original articles with complete proofs. Besides short notes the journal publishes also surveys as well as some issues are focusing on a theme of current interest.
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