Properties of the division topology on the set of positive integers (Q2800158)

The division topology on the set of positive integers introduced by \textit{G. B. Rizza} [Riv. Mat. Univ. Parma, V. Ser. 2, 179--185 (1993; Zbl 0834.11006)] is the right topology associated with the division order. A basis of this topology is \(\mathcal{B}=\{a\mathbb{N} \mid a\in \mathbb{N}\}\). This is a \(T_0\)-Alexandroff topology (i.e., a Kolmogoroff topology in which any intersection of open sets is open).NEWLINENEWLINEThe author of the paper under review gives some properties of the division topology; with a special focus on the closure of arithmetic progressions, compact sets and connected sets. If \(a=p_1^{\alpha_1} p_2^{\alpha_2}\ldots p_k^{\alpha_k}\) is the prime factorization of \(a\), then the author shows that the closure of the arithmetic progression \(\{an+b:n\in \mathbb{N}\}\) is the intersection of the closures \(\overline{\{ p_i^{\alpha_i}n+b:n\in \mathbb{N}\}}\). The paper is full of nice results with elementary elegant proofs. The last section of the paper deals with the Darboux property (a function \(f: X \to Y\) is said to have the Darboux property if the image by \(f\) of a connected set of \(X\) is a connected set of \(Y\)). The author shows that for the division topology, continuity and Darboux property are equivalent.NEWLINENEWLINEComments of the reviewer: It is not difficult to recognize connected and compact sets of an Alexandroff space. Let \((X,\mathcal{T})\) be an Alexandroff space with specialization quasi-order \(\leq\) (where \(\leq\) is defined by \(x\leq y\) if and only if \(x\in \overline{\{y\}})\). According to \textit{E. Bouacida} et al. [Boll. Unione Mat. Ital., VII. Ser., B 10, No. 2, 417--439 (1996; Zbl 0865.54032)]:NEWLINENEWLINE- A subset \(A\) of \(X\) is compact if and only if \(Min (A) \) is finite (where \(Min (A) \) stands for the set of minimal elements of \(A\).NEWLINENEWLINE- The set \(A\) is connected if and only if for all \(x,y\in A\), there exist \(x_0, x_1, \ldots , x_n\in A\) such \(x_0=x, x_n=y\) and and \(x_i\leq x_{i+1}\) or \(x_{i+1}\leq x_{i}\), for all \(i\).NEWLINENEWLINEThese comments make most of the results of Section 7 straightforward.