In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ℕ which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of the topological spaces (ℕ, T) and (ℕ, T′).
@article{bwmeta1.element.doi-10_2478_s11533-013-0210-3,
author = {Paulina Szczuka},
title = {Connections between connected topological spaces on the set of positive integers},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {876-881},
zbl = {1331.54021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0210-3}
}
Paulina Szczuka. Connections between connected topological spaces on the set of positive integers. Open Mathematics, Tome 11 (2013) pp. 876-881. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0210-3/
[1] Arkhangelskii A.V., Pontryagin L.S. (Eds.), General Topology, I, Encyclopaedia Math. Sci., 17, Springer, Berlin, 1990
[2] Brown M., A countable connected Hausdorff space, In: Cohen L.M., The April Meeting in New York, Bull. Amer. Math. Soc., 1953, 59(4), 367
[3] Engelking R., General Topology, Mathematical Monographs, 60, PWN, Warsaw, 1977
[4] Furstenberg H., On the infinitude of primes, Amer. Math. Monthly, 1955, 62(5), 353 http://dx.doi.org/10.2307/2307043 | Zbl 1229.11009
[5] Golomb S.W., A connected topology for the integers, Amer. Math. Monthly, 1959, 66(8), 663–665 http://dx.doi.org/10.2307/2309340 | Zbl 0202.33001
[6] Kirch A.M., A countable, connected, locally connected Hausdorff space, Amer. Math. Monthly, 1969, 76(2), 169–171 http://dx.doi.org/10.2307/2317265 | Zbl 0174.25602
[7] LeVeque W.J., Topics in Number Theory, I–II, Dover, Mineola, 2002 | Zbl 1009.11001
[8] Rizza G.B., A topology for the set of nonnegative integers, Riv. Mat. Univ. Parma, 1993, 2, 179–185 | Zbl 0834.11006
[9] Szczuka P., The connectedness of arithmetic progressions in Furstenberg’s, Golomb’s, and Kirch’s topologies, Demonstratio Math., 2010, 43(4), 899–909 | Zbl 1303.11021