On some relations between ideals of nowhere dense sets in topologies on positive integers

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 IG\(IK∪IF)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}_G {\setminus } ({{\mathcal {I}}}_K \cup {{\mathcal {I}}}_F)$$\end{document}. 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.


Preliminaries
The symbol N and Primes denote the set of positive integers and the set of primes, respectively. Let square greater than 1). Following [10], for all a, b ∈ N, the symbol {an + b} stands for the infinite arithmetic progressions: {an + b} = {a · n + b : n ∈ N ∪ {0}}. Moreover, define an abbreviation: {an} = {an + a}. Let us say that a family F ⊆ [N] N has the splitting property if for any F ∈ F one can find F 1 , Let us define various topologies on N: Furstenberg's topology was first defined in [1] to present a topological proof of the existence of infinitely many prime numbers. This topology is metrizable, zero-dimensional and totally disconnected. Furstenberg's topology was originally defined on the set of integers but in this paper, in order to make our presentation more unified, we trim this topology to N. Notice that the trimmed Furstenberg's topology is also metrizable, zero-dimensional and totally disconnected. Golomb's topology was defined in [2] to present a similar proof. Notice that Golomb's topology is Hausdorff but not regular. Kirch's topology was defined in [3] and this topology is again Hausdorff but not regular. Kirch's topology is weaker than Golomb's topology and it is locally connected, as opposed to Golomb's topology. Moreover, Rizza in [6] introduced the division topology. In [9] the author defined the common division topology T on N, stronger than the division topology T . Both topologies T and T are T 0 , they are not T 1 , and they are connected-however, the common division topology T is not locally connected, as opposed to the division topology T . Let us notice that all these topologies have recently been studied by P. Szyszkowska née Szczuka, e.g., in [7,8,10].
An ideal on N is a family of subsets of N closed under taking finite unions and subsets of its elements. We assume that an ideal is proper and contains all finite sets. Obviously, in any T 1 topology without isolated points, the nowhere dense sets form an ideal. For each topology defined above, consider the ideal of nowhere dense sets: I G , I K , I F , I S , I R in Golomb's, Kirch's, Furstenberg's, the common division, and Rizza's topology, respectively.

Results
In [5] the authors examined properties of the ideals I G , I K , and I F and they asked if it is true that I G \(I K ∪ I F ) = ∅ ([5, Problem 2.10]). It turns out that it is true. In the proof of the proposition below we present a simple solution of this problem.
Proof We construct the set Ex as in the proof of [5, Theorem 2.9]. Namely, define Let {C k : k ∈ N} be an enumeration of C. The set Ex = {x k : k ∈ N} is constructed as follows: for every k ∈ N we pick x k such that x k ∈ C k and x k ∈ {2 k n + 1}. In the proof of [5,Theorem 2.9] it was shown that Ex ∈ (I G ∩ I F )\I K . Now let By [5, Example 2.6], {2n} ∈ I G \I F . So, X ∈ I G as the sum of two nowhere dense sets in Golomb's topology, but X / The diagram below is described in [5]. Our recent example (Proposition 2.1) can be seen in this diagram describing the relations between the ideals I K , I G and I F . Hence, we finally completed the diagram.
In the next part of this paper we will mainly focus on the relationships between the ideals I R , I S and I F . However, some relations between all five ideals will also be examined. At first, we present a proposition showing that Furstenberg's, Rizza's and the common division ideal are not disjoint.

Proposition 2.2 There is an infinite set in
Moreover, . The next propositions will show that the ideals I R , I S and I F differ significantly.
Indeed, if U ∈ T \{∅}, then we can find {an + b} ⊆ U such that (a) ⊆ (b). Let us consider the following cases: Moreover, This follows from {2n+1}∈D and {2n+1} ∈ D . Therefore {2n+1} / ∈ I G and {2n+1} / ∈ I K .  To prove the next proposition we will need the characterization of closed sets in the common division topology.

Remark 2.8
There is a characterization of the closure of a set in the division topology, namely: cl T (A) = a∈A D(a) (see [6]).
Let us formulate an analogous characterization for the common division topology: Observe that x ∈ B is equivalent to ∀ k∈N ∃ a k ∈A a k ∈ {x k n + x} which in turn is equivalent to ∀ k∈N A ∩ {x k n + x} = ∅.
If x / ∈ B, then ∃ k∈N A ∩ {x k n + x} = ∅. Since x ∈ {x k n + x} and {x k n + x} ∈ T , we have x / ∈ cl T (A). Now suppose x ∈ B. If x = 1, then by [9,Proposition 3.1] x ∈ cl T (A). So, we can suppose x = 1. Let us choose U ∈ T such that x ∈ U . By [10, Lemma 3.1], there exists 1 1 · · · p α m m be a prime factor decomposition of x. Since (c) ⊆ (x), without loss of generality we may assume that c = p , so x ∈ cl T (A 2 ). Observe that A 2 ⊆ {x 2 n}, and since x ∈ cl T (A 2 ), by Corollary 2.10, we deduce A 2 ∩ {x 2 n + x} = ∅, which is impossible, since {x 2 n} ∩ {x 2 n + x} = ∅.
Next, int T (cl T (A)) = int T (A) = ∅, which yields A ∈ I S .
Moreover, we obtain This follows from [5, Example 2.1], where it was shown that A = {n!: n ∈ N} ∈ I G ∩ I K .
Proof Let A = {n!: n ∈ N} and X = A ∪ {2n + 1}. Then X ∈ I S , since A ∈ I S and {2n + 1} ∈ I S . Observe that X / ∈ I R since A / ∈ I R and X / ∈ I F since {2n + 1} / ∈ I F (cf. the proofs of Proposition 2.12 and Proposition 2.3).

Moreover,
This follows from {2n + 1} / ∈ I K ∪ I G . All relations among the ideals I S , I R and I F proven in this article can be seen in the following diagram: Finally, note that if the answer to Problem 2.14 is positive, then the answer to Problem 2.15 will also be positive.
Let us end the article with a result about the splitting property of the common division topology.
Namely, by Proposition 1.1 from [5] we know that any base for any Hausdorff topology without isolated points has the splitting property. However, the division and the common division topology is not Hausdorff, therefore it is natural to ask whether these two topologies have the splitting property. We obtain Then in both cases F 1 , F 2 ∈ B T , F 1 ∩ F 2 = ∅ and F 1 ∪ F 2 ⊆ {an + b}.
On the other hand, the family B T does not have the splitting property, since ab ∈ {an} ∩ {bn}.
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