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Connections between connected topological spaces on the set of positive integers

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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′).
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Bibliografia
  • [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
  • [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
  • [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
  • [7] LeVeque W.J., Topics in Number Theory, I–II, Dover, Mineola, 2002
  • [8] Rizza G.B., A topology for the set of nonnegative integers, Riv. Mat. Univ. Parma, 1993, 2, 179–185
  • [9] Szczuka P., The connectedness of arithmetic progressions in Furstenberg’s, Golomb’s, and Kirch’s topologies, Demonstratio Math., 2010, 43(4), 899–909
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bwmeta1.element.doi-10_2478_s11533-013-0210-3
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