Connections between connected topological spaces on the set of positive integers

Paulina Szczuka

Open Mathematics (2013)

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

Abstract

top
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′).

How to cite

top

Paulina 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

top
  1. [1] Arkhangelskii A.V., Pontryagin L.S. (Eds.), General Topology, I, Encyclopaedia Math. Sci., 17, Springer, Berlin, 1990 
  2. [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. [3] Engelking R., General Topology, Mathematical Monographs, 60, PWN, Warsaw, 1977 
  4. [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. [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. [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. [7] LeVeque W.J., Topics in Number Theory, I–II, Dover, Mineola, 2002 Zbl1009.11001
  8. [8] Rizza G.B., A topology for the set of nonnegative integers, Riv. Mat. Univ. Parma, 1993, 2, 179–185 Zbl0834.11006
  9. [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

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.