The Darboux property for polynomials in Golomb's and Kirch's topologies

• Autorzy: Szczuka P.
• Języki publikacji: EN

Abstrakty

EN

In this paper, 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.

Słowa kluczowe

EN

Golomb's topology; Kirch's topology; continuity; Darboux property; polynomial; arithmetic progression

PL

topologia Golomba; topologia Kircha; ciągłość; własność Darboux; wielomian; progresja arytmetyczna

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2013
• Tom: Vol. 46, nr 2
• Strony: 429--435
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Szczuka P.

• Kazimierz Wielki University, Pl. Weyssenhoffa 11, 85–072 Bydgoszcz, Poland

Bibliografia

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• [2] M. Brown, A countable connected Hausdorff space, Bull. Amer. Math. Soc. 59 (1953), 367.
• [3] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
• [4] H. Furstenberg, On the infinitude of primes, Amer. Math. Monthly 62 (1955), 353.
• [5] S. Golomb, A connected topology for the integers, Amer. Math. Monthly 66 (1959), 663–665.
• [6] A. M. Kirch, A countable, connected, locally connected Hausdorff space, Amer. Math. Monthly 76 (1969), 169–171.
• [7] W. J. LeVeque, Topics in Number Theory, Vol. I, II, Dower Publications Inc., New York, 2002.
• [8] P. Szczuka, The connectedness of arithmetic progressions in Furstenberg’s, Golomb’s, and Kirch’s topologies, Demonstratio Math. 43(4) (2010), 899–909.