Previous |  Up |  Next

Article

Keywords:
arithmetic progression; common division topology; totally Brown space; totally separated space
Summary:
A topological space $X$ is totally Brown if for each $n \in \mathbb{N} \setminus \{1\}$ and every nonempty open subsets $U_1,U_2,\ldots,U_n$ of $X$ we have ${\rm cl}_X(U_1) \cap {\rm cl}_X(U_2) \cap \cdots \cap{\rm cl}_X(U_n) \ne \emptyset$. Totally Brown spaces are connected. In this paper we consider a topology $\tau_S$ on the set $\mathbb{N}$ of natural numbers. We then present properties of the topological space $(\mathbb{N},\tau_S)$, some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by P. Szyszkowska and M. Szyszkowski in 2018.
References:
[1] Alberto-Domínguez J. C., Acosta G., Delgadillo-Piñón G.: Totally Brown subsets of the Golomb space and the Kirch space. Comment. Math. Univ. Carolin. 63 (2022), no. 2, 189–219. MR 4506132
[2] Apostol T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York, 1976. MR 0434929
[3] Aull C. E., Thron W. J.: Separation axioms between $T_0$ and $T_1$. Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 26–37. DOI 10.1016/S1385-7258(62)50003-6 | MR 0138082
[4] Dontchev J.: On superconnected spaces. Serdica 20 (1994), no. 3–4, 345–350. MR 1333356
[5] Dunham W.: $T_{1/2}$-spaces. Kyungpook Math. J. 17 (1977), no. 2, 161–169. MR 0470934
[6] Engelking R.: General Topology. Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[7] Fine B., Rosenberger G.: Number Theory. An Introduction via the Density of Primes. Birkhäuser/Springer, Cham, 2016. MR 3559913
[8] Golomb S. W.: A connected topology for the integers. Amer. Math. Monthly 66 (1959), 663–665. DOI 10.1080/00029890.1959.11989385 | MR 0107622
[9] Golomb S. W.: Arithmetica topologica. General Topology and Its Relations to Modern Analysis and Algebra, Proc. Sympos., Praha, 1961, Academic Press, New York, Publ. House Czech. Acad. Sci., Praha, 1962, pages 179–186 (Italian). MR 0154249
[10] Jha M. N.: Separation axioms between $T_0$ and $T_1$. Progr. Math. (Allahabad) 11 (1977), no. 1–2, 1–4. MR 0458365
[11] Kirch A. M.: A countable, connected, locally connected Hausdorff space. Amer. Math. Monthly 76 (1969), 169–171. DOI 10.1080/00029890.1969.12000163 | MR 0239563
[12] Levine N.: Generalized closed sets in topology. Rend. Circ. Mat. Palermo (2) 19 (1970), 89–96. DOI 10.1007/BF02843888 | MR 0305341
[13] Nanda S., Panda H. K.: The fundamental group of principal superconnected spaces. Rend. Mat. (6) 9 (1976), no. 4, 657–664. MR 0434295
[14] Steen L. A., Seebach J. A., Jr.: Counterexamples in Topology. Dover Publications, Mineola, New York, 1995. MR 1382863 | Zbl 0386.54001
[15] Szczuka P.: Connections between connected topological spaces on the set of positive integers. Cent. Eur. J. Math. 11 (2013), no. 5, 876–881. MR 3032336
[16] Szczuka P.: The closures of arithmetic progressions in the common division topology on the set of positive integers. Cent. Eur. J. Math. 12 (2014), no. 7, 1008–1014. MR 3188461
[17] Szczuka P.: Properties of the division topology on the set of positive integers. Int. J. Number Theory 12 (2016), no. 3, 775–785. DOI 10.1142/S1793042116500500 | MR 3477420
[18] Szyszkowska P., Szyszkowski M.: Properties of the common division topology on the set of positive integers. J. Ramanujan Math. Soc. 33 (2018), no. 1, 91–98. MR 3772612
Partner of
EuDML logo