Connections between connected topological spaces on the set of positive integers

• Autorzy: Paulina Szczuka
• Języki publikacji: EN

Abstrakty

EN

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′).

Słowa kluczowe

EN

Division topology; Connectedness; Local connectedness; Arithmetic progression

Kategorie tematyczne

• 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
• 54D05: Connected and locally connected spaces (general aspects)
• 11B25: Arithmetic progressions

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 5
• Strony: 876-881

Daty

wydano

2013-05-01

online

2013-03-14

Twórcy

autor

Paulina Szczuka

• Kazimierz Wielki University

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

Identyfikatory

DOI

10.2478/s11533-013-0210-3