Research and Innovation Articles
The Golomb space and its non connectedness "im kleinen"
Published 2018-03-06
Keywords
- Arithmetic progression,
- connectedness,
- Golomb topology,
- local connectedness
How to Cite
Alberto-Domínguez, J. del C., Acosta, G., Delgadillo-Piñón, G., & Madriz-Mendoza, M. (2018). The Golomb space and its non connectedness "im kleinen". Revista Integración, Temas De matemáticas, 35(2), 189–213. https://doi.org/10.18273/revint.v35n2-2017004
Abstract
In the present paper we study Brown spaces which are connected and not completely Hausdorff. Using arithmetic progressions, we construct a base BG for a topology τG on N, and show that (N, τG), called the Golomb space is a Brown space. We also show that some elements of BG are Brown spaces, while others are totally separated. We write some consequences of such result. For example, the space (N, τG) is not connected "im kleinen" at each of its points. This generalizes a result proved by Kirch in 1969. We also present a simpler proof of a result given by Szczuka in 2010.
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References
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[2] Brown M., "A countable connected Hausdorff space", in The April Meeting in New York, Bull. Amer. Math. Soc. 59 (1953), No. 4, 367.
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[10] Margalef Roig J., Outerelo Domínguez E. and Pinilla Ferrando J.L., Topología, Editorial Alhambra, Madrid, 1979.
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[13] Szczuka P., "The connnectedness of arithmetic progressions in Furstenberg's, Golomb's and Kirch's topologies", Demonstratio Math. 43 (2010), No. 4, 899-909.
[14] Szczuka P., "Connections between connected topological spaces on the set of positive integers", Cent. Eur. J. Math. 11 (2013), No. 5, 876-881.
[15] Szczuka P., "Regular open arithmetic progressions in connected topological spaces on the set of positive integers", Glas. Mat. Ser. III 49(69)(2014), No. 1, 13-23.
[16] Szczuka P., "The closures of arithmetic progressions in the common division topology on the set of positive integers", Cent, Eur. J. Math. 12 (2014), No. 7, 1008-1014.
[17] Szczuka P., "The closures of arithmetic progressions in Kirch's topology on the set of positive
integers", Int. J. Number Theory 11 (2015), No. 3, 673-682.
[18] Szczuka P., "Properties of the division topology on the set of positive integers", Int. J. Number Theory 12 (2016), No. 3, 775-785.
[19] Willard S., General Topology, Reprint of the 1970 Original Addison-Wesley, Rending, MA. Dover Publications, Inc., Mineola, New York, 2004.
[2] Brown M., "A countable connected Hausdorff space", in The April Meeting in New York, Bull. Amer. Math. Soc. 59 (1953), No. 4, 367.
[3] Clark P.L., Lebowitz-Lockard N. and Pollack P., "A note on Golomb topologies", Preprint.
[4] Engelking R., General Topology, Revised and Completed Edition, Sigma Series in Pure Mathematics Vol.6. Heldermann Verlag, Berlin, 1989.
[5] Fine B. and Rosenberger G., Number Theory. An Introduction via the Density of Primes, Second Edition, Birkhäuser, 2016.
[6] Furstenberg H., "On the infnitude of primes", Amer. Math. Monthly 62 (1955), No. 5, 353.
[7] Golomb S.W., "A connected topology for the integers", Amer. Math. Monthly 66 (1959), No 8, 663-665.
[8] Golomb S.W., "Arithmetica topologica", General Topology and its Relations to Modern Analysis and Algebra, Proceedings of the Symposium held in Prague in September 1961. Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1962, 179-186.
[9] Kirch A.M., "A countable, connected, locally connected Hausdorff space", Amer. Math. Monthly 76 (1969), No. 2, 169-171.
[10] Margalef Roig J., Outerelo Domínguez E. and Pinilla Ferrando J.L., Topología, Editorial Alhambra, Madrid, 1979.
[11] Niven I., Zuckerman H.S. and Montgomery H.L., An Introduction to the Theory of Numbers, Fifth Edition, John Wiley and Sons, Inc., New York, Chichester, Brisbane, Toronto, Singapore, 1991.
[12] Steen L.A. and Seebach Jr. J.A., Counterexamples in Topology, Reprint of the Second (1978) Edition Original Springer-Verlag, New York. Dover Publications, Inc., Mineola, New York, 1995.
[13] Szczuka P., "The connnectedness of arithmetic progressions in Furstenberg's, Golomb's and Kirch's topologies", Demonstratio Math. 43 (2010), No. 4, 899-909.
[14] Szczuka P., "Connections between connected topological spaces on the set of positive integers", Cent. Eur. J. Math. 11 (2013), No. 5, 876-881.
[15] Szczuka P., "Regular open arithmetic progressions in connected topological spaces on the set of positive integers", Glas. Mat. Ser. III 49(69)(2014), No. 1, 13-23.
[16] Szczuka P., "The closures of arithmetic progressions in the common division topology on the set of positive integers", Cent, Eur. J. Math. 12 (2014), No. 7, 1008-1014.
[17] Szczuka P., "The closures of arithmetic progressions in Kirch's topology on the set of positive
integers", Int. J. Number Theory 11 (2015), No. 3, 673-682.
[18] Szczuka P., "Properties of the division topology on the set of positive integers", Int. J. Number Theory 12 (2016), No. 3, 775-785.
[19] Willard S., General Topology, Reprint of the 1970 Original Addison-Wesley, Rending, MA. Dover Publications, Inc., Mineola, New York, 2004.