zbMATH — the first resource for mathematics

The Darboux property for polynomials in Golomb’s and Kirch’s topologies. (English) Zbl 1303.54004
Summary: We present conditions which are equivalent to the Darboux property for non-constant polynomials in Golomb’s and Kirch’s topologies on the set of positive integers.

54C10 Special maps on topological spaces (open, closed, perfect, etc.)
11C08 Polynomials in number theory
54C05 Continuous maps
Full Text: DOI
[1] [1] K. A. Broughan, Adic topologies for the rational integers, Canad. J. Math. 55 (2003), 711-723. · Zbl 1060.11004
[2] [2] M. Brown, A countable connected Hausdorff space, Bull. Amer. Math. Soc. 59 (1953), 367. · Zbl 0051.13902
[3] [3] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
[4] [4] H. Furstenberg, On the infinitude of primes, Amer. Math. Monthly 62 (1955), 353.
[5] [5] S. Golomb, A connected topology for the integers, Amer. Math. Monthly 66 (1959), 663-665. · Zbl 0202.33001
[6] [6] A. M. Kirch, A countable, connected, locally connected Hausdorff space, Amer. Math. Monthly 76 (1969), 169-171. · Zbl 0174.25602
[7] [7] W. J. LeVeque, Topics in Number Theory, Vol. I, II, Dower Publications Inc., New York, 2002. · Zbl 1009.11001
[8] [8] P. Szczuka, The connectedness of arithmetic progressions in Furstenberg’s, Golomb’s, and Kirch’s topologies, Demonstratio Math. 43(4) (2010), 899-909. · Zbl 1303.11021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.