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

Open Mathematics (2014)

- Volume: 12, Issue: 7, page 1008-1014
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topPaulina Szczuka. "The closures of arithmetic progressions in the common division topology on the set of positive integers." Open Mathematics 12.7 (2014): 1008-1014. <http://eudml.org/doc/269569>.

@article{PaulinaSzczuka2014,

abstract = {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.},

author = {Paulina Szczuka},

journal = {Open Mathematics},

keywords = {The common division topology; Closures; Arithmetic progressions; the common division topology; closures; arithmetic progressions},

language = {eng},

number = {7},

pages = {1008-1014},

title = {The closures of arithmetic progressions in the common division topology on the set of positive integers},

url = {http://eudml.org/doc/269569},

volume = {12},

year = {2014},

}

TY - JOUR

AU - Paulina Szczuka

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

JO - Open Mathematics

PY - 2014

VL - 12

IS - 7

SP - 1008

EP - 1014

AB - 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.

LA - eng

KW - The common division topology; Closures; Arithmetic progressions; the common division topology; closures; arithmetic progressions

UR - http://eudml.org/doc/269569

ER -

## References

top- [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 Zbl1229.11009
- [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 Zbl0202.33001
- [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 Zbl0174.25602
- [6] LeVeque W.J., Topics in Number Theory, I-II, Dover, Mineola, 2002 Zbl1009.11001
- [7] Rizza G.B., A topology for the set of nonnegatlve integers, Riv. Mat. Univ. Parma, 1993, 2, 179–185 Zbl0834.11006
- [8] Szczuka P., The connectedness of arithmetic progressions in Furstenberg’s, Golomb’s, and Kirch’s topologies, Demonstratio Math., 2010, 43(4), 899–909 Zbl1303.11021
- [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 Zbl1331.54021
- [10] Szczuka P., Regular open arithmetic progressions in connected topological spaces on the set of positive integers, Glas. Mat. (in press) Zbl06374862

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.