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

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