The closures of arithmetic progressions in the common division topology on the set of positive integers

• Autorzy: Paulina Szczuka
• Języki publikacji: EN

Abstrakty

EN

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.

Słowa kluczowe

EN

The common division topology; Closures; Arithmetic progressions

Kategorie tematyczne

• 11A41: Primes
• 54A05: Topological spaces and generalizations (closure spaces, etc.)
• 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
• 11B25: Arithmetic progressions

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 7
• Strony: 1008-1014

Daty

wydano

2014-07-01

online

2014-04-03

Twórcy

autor

Paulina Szczuka

• Kazimierz Wielki University

Bibliografia

• [1] 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
• [2] Furstenberg H., On the Infinitude of primes, Amer. Math. Monthly, 1955, 62(5), 353 http://dx.doi.org/10.2307/2307043
• [3] Golomb S.W., A connected topology for the integers, Amer. Math. Monthly, 1959, 66(8), 663–665 http://dx.doi.org/10.2307/2309340
• [4] Kelley J.L., General Topology, Grad. Texts in Math., 27, Springer, New York-Berlin, 1975
• [5] 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
• [6] LeVeque W.J., Topics in Number Theory, I-II, Dover, Mineola, 2002
• [7] Rizza G.B., A topology for the set of nonnegatlve integers, Riv. Mat. Univ. Parma, 1993, 2, 179–185
• [8] Szczuka P., The connectedness of arithmetic progressions in Furstenberg’s, Golomb’s, and Kirch’s topologies, Demonstratio Math., 2010, 43(4), 899–909
• [9] Szczuka P., Connections between connected topological spaces on the set of positive integers, Cent. Eur. J. Math., 2013, 11(5), 876–881 http://dx.doi.org/10.2478/s11533-013-0210-3
• [10] Szczuka P., Regular open arithmetic progressions in connected topological spaces on the set of positive integers, Glas. Mat. (in press)

Identyfikatory

DOI

10.2478/s11533-013-0394-6