On some relations between ideals of nowhere dense sets in topologies on positive integers
Andrzej Nowik, Paulina Szyszkowska · Periodica Mathematica Hungarica · 2021
Abstract 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 $${{\mathcal {I}}}_G {\setminus } ({{\mathcal {I}}}_K \cup {{\mathcal {I}}}_F)$$ I G \ ( I K ∪ 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.