The connectedness of arithmetic progressions in Furstenberg’s, Golomb’s, and Kirch’s topologies. (English) Zbl 1303.11021

Szczuka, Paulina

Demonstr. Math. 43, No. 4, 899-909 (2010).

The paper contains results on connectedness and local connectedness of arithmetic progressions in topologies introduced by H. Furstenberg [Am. Math. Mon. 62, No. 5, 353 (1955; Zbl 1229.11009)], S. W. Golomb [Am. Math. Mon. 66, 663–665 (1959; Zbl 0202.33001)] or A. M. Kirch [Am. Math. Mon. 76, 169–171 (1969; Zbl 0174.25602)] on the set of integers or on the set of positive integers, respectively. Since Furstenberg and Golomb topologies were introduced as a tool to prove basic infinitude assertions on the infinitude of prime numbers, the author also investigates some related aspects of the set of primes in these topologies. (For other algebro-topological connections with Furstenberg and Golomb topologies consult reviewer previous or forthcoming papers, e.g. [Expo. Math. 15, No. 2, 131–148 (1997; Zbl 0883.11043)], [Int. J. Number Theory 8, No. 3, 823–830 (2012; Zbl 1290.11012)]).

Reviewer: Štefan Porubský (Praha)

MSC:

11B25	Arithmetic progressions
11A41	Primes
54D05	Connected and locally connected spaces (general aspects)

Keywords:

Furstenberg topology; Golomb topology; Kirch topology; connectedness; local connectedness, arithmetic progressions; prime numbers