Article
MSC:
11A41,
11B05,
11B25,
54A05,
54D05,
54D10 | MR 4717507 | Zbl 07830514 | DOI: 10.14712/1213-7243.2023.029
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Keywords:
aposyndesis; arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space
aposyndesis; arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space
Summary:
We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions $P(a,b)$ with the property that every prime number that divides $a$ also divides $b$, it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are aposyndetic in the Golomb space.
References:
[1] Alberto-Domínguez J. D. 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] Alberto-Domínguez J. D. C., Acosta G., Madriz-Mendoza M.: The common division topology on $\mathbb{N}$. Comment. Math. Univ. Carolin. 63 (2022), no. 3, 329–349. MR 4542793
[3] Banakh T., Mioduszewski J., Turek S.: On continuous self-maps and homeomorphisms of the Golomb space. Comment. Math. Univ. Carolin. 59 (2018), no. 4, 423–442. MR 3914710
[4] Engelking R.: General Topology. Sigma Ser. Pure Math., 6, Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[5] Golomb S. W.: A connected topology for the integers. Amer. Math. Monthly 66 (1959), 663–665. DOI 10.1080/00029890.1959.11989385 | MR 0107622
[6] Golomb S. W.: Arithmetica topologica. General Topology and Its Relations to Modern Analysis and Algebra, Proc. Sympos., Prague, 1961, Academic Press, New York, 1961, pages 179–186 (Italian). MR 0154249
[7] 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
[8] Steen L. A., Seebach J. A., Jr.: Counterexamples in Topology. Dover Publications, Mineola, New York, 1995. MR 1382863 | Zbl 0386.54001
[9] Szczuka P.: The connectedness of arithmetic progressions in Furstenberg's, Golomb's, and Kirch's topologies. Demonstratio Math. 43 (2010), no. 4, 899–909. DOI 10.1515/dema-2010-0416 | MR 2761648
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