We examine the ideals of nowhere dense sets in three topologies on the set of positive integers, namely Furstenberg’s, Rizza’s and the common division topology. We mainly concentrate on inclusions between these ideals, we present a diagram showing these and we explore all possible inclusions between them. We present a formula for the closure of a set in the common division topology. We answer a question posed by Kwela and Nowik (Topol Appl. 248:149–163, 2018) by constructing a set in \({{\mathcal {I}}}_G {\setminus } ({{\mathcal {I}}}_K \cup {{\mathcal {I}}}_F)\). Therefore, the main diagram of comparison between the ideals of nowhere dense sets in various topologies from the article by M. Kwela and A. Nowik is completed.

我们研究了正整数集上的三种拓扑中无处稠密集的理想，即 Furstenberg 拓扑、Rizza 拓扑和公分拓扑。我们主要关注这些理想之间的包含，我们展示了一个图表，我们探索了它们之间所有可能的包含。我们提出了公分拓扑中集合的闭包公式。我们通过在\({{\mathcal {I}}}_G {\setminus } ({{\mathcal {I} }}_K \cup {{\mathcal {I}}}_F)\)。因此，完成了 M. Kwela 和 A. Nowik 的文章中各种拓扑中无处稠密集理想比较的主要图。