Properties of arithmetics progressions in increasing sequence of \(T_0\)-topologies on the set of positive integers (Q6545216)

The sets of integers, positive integers, non-negative integers, and primes are denoted by \(\mathbb{Z}, \mathbb{N}, \mathbb{N}_0,\) and \(\mathcal{P}\), respectively.\N\NLet \(A\) be a subset of a topological space, we denote by \(\overline{A}\) and \(A^\circ\) the closure and interior of \(A\), respectively.\N\NIf \(a \in \mathbb{N}\), then \(\Theta(a)\) represents the set of all prime factors of \(a\), and \(l_p(a)\) denotes the largest integer such that \(p^{l_p(a)}\) divides \(a\).\N\NFor \(a, b \in \mathbb{N}\), we define \(\{an + b\} := a \cdot \mathbb{N}_0 + b\) and \(\{az + b\} := a \cdot \mathbb{Z} + b\).\N\NIn 1955, \textit{H. Furstenberg} [Am. Math. Mon. 62, No. 5, 353 (1955; Zbl 1229.11009)] introduced a topology \(\mathcal{T}_F\) on the integers with a basis consisting of all arithmetic progressions \(\{an + b\}\) and provided a topological proof of the infinitude of primes. In 1959, \textit{S. W. Golomb} [Am. Math. Mon. 66, 663--665 (1959; Zbl 0202.33001)] offered a similar proof using a topology \(\mathcal{D}\), defined in 1953 by \textit{M. Brown} [``A countable connected Hausdorff space'', Bull. Am. Math. Soc. 59, 367 (1953)], on \(\mathbb{N}\) with the basis\N\[\N\mathcal{B}_G = \{\{an + b\} : \gcd(a, b) = 1\}.\N\]\NTen years later, \textit{A. M. Kirch} [Am. Math. Mon. 76, 169--171 (1969; Zbl 0174.25602)] introduced a weaker topology \(\mathcal{D}'\) on \(\mathbb{N}\), with basis\N\[\N\mathcal{B}_K = \{\{an + b\} : \gcd(a, b) = 1, \text{\(a\) is square-free}\}.\N\]\N\NIn 2022, \textit{P. Szyszkowska} [Topology Appl. 314, Article ID 108096, 8 p. (2022; Zbl 1495.54015)] defined an increasing sequence \(\{\mathcal{D}_m\}\) of connected and locally connected Hausdorff topologies on \(\mathbb{N}\) such that the sum of all topologies \(\mathcal{D}_m\) is Golomb's topology, and \(\mathcal{D}_1\) coincides with Kirch's topology. The basis of each topology \(\mathcal{D}_m\) consists of all arithmetic progressions \(\{an + b\}\) that are open in Golomb's topology, where the largest power of each prime factor dividing \(a\) does not exceed \(m\).\N\NMoreover, if \(m \in \mathbb{N} \setminus \{1\}\), then the space \((\mathbb{N}, \mathcal{D}_m)\) is semiregular [\textit{P. Szyszkowska}, Topology Appl. 317, Article ID 108188, 8 p. (2022; Zbl 1504.54016)], whereas \((\mathbb{N}, \mathcal{D}_1)\), with Kirch's topology, is not semiregular [\textit{P. Szczuka}, Glas. Mat., III. Ser. 49, No. 1, 13--23 (2014; Zbl 1348.11010)].\N\NThe division topology \(\mathcal{T}'\) on \(\mathbb{N}\) was introduced in 1993 by \textit{G. B. Rizza} [Riv. Mat. Univ. Parma, V. Ser. 2, 179--185 (1993; Zbl 0834.11006)], where the Kuratowski closure is specified as follows: for \(X \subseteq \mathbb{N}\),\N\[\N\overline{X} = \bigcup_{x \in X} D(x), \quad \text{where } D(x) = \{y \in \mathbb{N} : y \mid x\}.\N\]\NThe family \(\mathcal{B}' = \{\{an\}\}\) forms a basis for this topology. In [\textit{P. Szczuka}, Cent. Eur. J. Math. 11, No. 5, 876--881 (2013; Zbl 1331.54021)], a stronger topology \(\mathcal{T}\), termed the common division topology, was defined with a basis\N\[\N\mathcal{B} = \{\{an + b\} : \Theta(a) \subseteq \Theta(b)\}.\N\]\N\NRecently, \textit{D. Krasiński} and \textit{P. Szyszkowska} [Topology Appl. 341, Article ID 108732, 9 p. (2024; Zbl 1537.54016)] introduced an increasing sequence \(\{\mathcal{T}_m\}\) of connected and locally connected \(T_0\)-topologies on \(\mathbb{N}\), such that the sum of all \(\mathcal{T}_m\) is the common division topology. Each topology \(\mathcal{T}_m\) has a basis consisting of arithmetic progressions \(\{an + b\}\) that are open in the common division topology and where the largest power of each prime factor dividing \(a\) does not exceed \(m\).\N\NThis paper investigates the topologies \(\mathcal{T}_m\) with a focus on the properties of arithmetic progressions.\N\NThe authors prove the following result: Let \(a = p_1^{\alpha_1} \cdots p_k^{\alpha_k}\) be the prime factorization of \(a\). Then, for every \(m \in \mathbb{N}\), the \(\mathcal{T}_m\)-topological closure of \(\{an + b\}\) is\N\[\N\overline{\{an + b\}} = \mathbb{N} \cap \bigcap_{i=1}^k \left( \{p_i^{\beta_i}z + b\} \cup (\mathbb{N} \setminus \{p_i n\}) \right),\N\]\Nwhere \(\beta_i = \min\{\alpha_i, m\}\) for each \(i \in \{1, \ldots, k\}\).\N\NLet \(X\) be a topological space. A subset \(A \subseteq X\) is called a regular open set if \(\overline{A}^\circ = A\). The space \(X\) is said to be semiregular if the collection of regular open sets forms a base for \(X\).\N\NThe authors demonstrate that, for every \(m \in \mathbb{N} \setminus \{1\}\), the space \((\mathbb{N}, \mathcal{T}_m)\) is semiregular.\N\NIt is worthnoting that the paper is well-written and well-structured.