{"title":"Pseudointersection numbers, ideal slaloms, topological spaces, and cardinal inequalities","authors":"Jaroslav Šupina","doi":"10.1007/s00153-022-00832-8","DOIUrl":null,"url":null,"abstract":"<div><p>We investigate several ideal versions of the pseudointersection number <span>\\(\\mathfrak {p}\\)</span>, ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant <span>\\(\\mathtt {cov}^*({\\mathcal I})\\)</span> has a crucial influence on the studied notions. For an invariant <span>\\(\\mathfrak {p}_\\mathrm {K}({\\mathcal J})\\)</span> introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant <span>\\(\\mathfrak {p}_\\mathrm {K}({\\mathcal I},{\\mathcal J})\\)</span> introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have </p><div><div><span>$$\\begin{aligned} \\min \\{\\mathfrak {p}_\\mathrm {K}({\\mathcal I}),\\mathtt {cov}^*({\\mathcal I})\\}=\\mathfrak {p},\\qquad \\min \\{\\mathfrak {p}_\\mathrm {K}({\\mathcal I},{\\mathcal J}),\\mathtt {cov}^*({\\mathcal J})\\}\\le \\mathtt {cov}^*({\\mathcal I}), \\end{aligned}$$</span></div></div><p>respectively. In addition to the first inequality, for a slalom invariant <span>\\(\\mathfrak {sl_e}({\\mathcal I},{\\mathcal J})\\)</span> introduced in Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that </p><div><div><span>$$\\begin{aligned} \\min \\{\\mathfrak {p}_\\mathrm {K}({\\mathcal I}),\\mathfrak {sl_e}({\\mathcal I},{\\mathcal J}),\\mathtt {cov}^*({\\mathcal J})\\}=\\mathfrak {p}. \\end{aligned}$$</span></div></div><p>Finally, we obtain a consistency that ideal versions of the Fréchet–Urysohn property and the strictly Fréchet–Urysohn property are distinguished.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00832-8.pdf","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-022-00832-8","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: 2
Abstract
We investigate several ideal versions of the pseudointersection number \(\mathfrak {p}\), ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant \(\mathtt {cov}^*({\mathcal I})\) has a crucial influence on the studied notions. For an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal J})\) introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J})\) introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have
respectively. In addition to the first inequality, for a slalom invariant \(\mathfrak {sl_e}({\mathcal I},{\mathcal J})\) introduced in Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that
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.