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

Paulina Szczuka

Open Mathematics (2014)

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

Abstract

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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.

How to cite

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Paulina 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

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  1. [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. [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. [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. [4] Kelley J.L., General Topology, Grad. Texts in Math., 27, Springer, New York-Berlin, 1975 
  5. [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. [6] LeVeque W.J., Topics in Number Theory, I-II, Dover, Mineola, 2002 Zbl1009.11001
  7. [7] Rizza G.B., A topology for the set of nonnegatlve integers, Riv. Mat. Univ. Parma, 1993, 2, 179–185 Zbl0834.11006
  8. [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. [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. [10] Szczuka P., Regular open arithmetic progressions in connected topological spaces on the set of positive integers, Glas. Mat. (in press) Zbl06374862

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