# Connections between connected topological spaces on the set of positive integers

Open Mathematics (2013)

- Volume: 11, Issue: 5, page 876-881
- ISSN: 2391-5455

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topPaulina Szczuka. "Connections between connected topological spaces on the set of positive integers." Open Mathematics 11.5 (2013): 876-881. <http://eudml.org/doc/269112>.

@article{PaulinaSzczuka2013,

abstract = {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′).},

author = {Paulina Szczuka},

journal = {Open Mathematics},

keywords = {Division topology; Connectedness; Local connectedness; Arithmetic progression; topology on the set of positive integers; divisors topology; connectedness; local connectedness; arithmetic progression},

language = {eng},

number = {5},

pages = {876-881},

title = {Connections between connected topological spaces on the set of positive integers},

url = {http://eudml.org/doc/269112},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Paulina Szczuka

TI - Connections between connected topological spaces on the set of positive integers

JO - Open Mathematics

PY - 2013

VL - 11

IS - 5

SP - 876

EP - 881

AB - 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′).

LA - eng

KW - Division topology; Connectedness; Local connectedness; Arithmetic progression; topology on the set of positive integers; divisors topology; connectedness; local connectedness; arithmetic progression

UR - http://eudml.org/doc/269112

ER -

## References

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- [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 Zbl1229.11009
- [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 Zbl0202.33001
- [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 Zbl0174.25602
- [7] LeVeque W.J., Topics in Number Theory, I–II, Dover, Mineola, 2002 Zbl1009.11001
- [8] Rizza G.B., A topology for the set of nonnegative integers, Riv. Mat. Univ. Parma, 1993, 2, 179–185 Zbl0834.11006
- [9] 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

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