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Golomb space; arithmetic progression; superconnected space; homeomorphism
The Golomb space ${\mathbb N}_\tau$ is the set ${\mathbb N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+ bn: n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb N}_\tau$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi$ of prime numbers is a dense metrizable subspace of ${\mathbb N}_\tau$, and each homeomorphism $h$ of ${\mathbb N}_\tau$ has the following properties: $h(1)=1$, $h(\Pi)=\Pi$, $\Pi_{h(x)}=h(\Pi_x)$, and $h(x^{{\mathbb N}})=h(x)^{\,\mathbb N}$ for all $x\in{\mathbb N}$. Here $x^{\mathbb N}:=\{x^n\colon n\in{\mathbb N}\}$ and $\Pi_x$ denotes the set of prime divisors of $x$.
[1] Apostol T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York, 1976. MR 0434929
[2] Artin E., Tate J.: Class Field Theory. Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, 1990. MR 1043169
[3] Banakh T.: Is the Golomb countable connected space topologically rigid?. available at
[4] Banakh T.: A simultaneous generalization of the Grunwald-Wang and Dirichlet theorems on primes. available at
[5] Banakh T.: Is the identity function a unique multiplicative homeomorphism of $ \mathbb N$?. available at
[6] Brown M.: A countable connected Hausdorff space. Bull. Amer. Math. Soc. 59 (1953), Abstract 423, 367.
[7] Clark P. L., Lebowitz-Lockard N., Pollack P.: A note on Golomb topologies. Quaest. Math. published online, available at DOI 10.2989/16073606.2018.1438533
[8] Dirichlet P. G. L.: Lectures on Number Theory. History of Mathematics, 16, American Mathematical Society, Providence, London, 1999. DOI 10.1090/hmath/016 | MR 1710911
[9] Engelking R.: General Topology. Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[10] Engelking R.: Theory of Dimensions Finite and Infinite. Sigma Series in Pure Mathematics, 10, Heldermann Verlag, Lemgo, 1995. MR 1363947 | Zbl 0872.54002
[11] Furstenberg H.: On the infinitude of primes. Amer. Math. Monthly 62 (1955), 353. DOI 10.2307/2307043 | MR 0068566
[12] Gauss C. F.: Disquisitiones Arithmeticae. Springer, New York, 1986. Zbl 1167.11001
[13] Golomb S. W.: A connected topology for the integers. Amer. Math. Monthly 66 (1959), 663–665. DOI 10.1080/00029890.1959.11989385 | MR 0107622
[14] Golomb S. W.: Arithmetica topologica. General Topology and Its Relations to Modern Analysis and Algebra, Proc. Symp., Prague, 1961, Academic Press, New York; Publ. House Czech. Acad. Sci., Praha (1962), 179–186. MR 0154249
[15] Jones G. A., Jones J. M.: Elementary Number Theory. Springer Undergraduate Mathematics Series, Springer, London, 1998. MR 1610533
[16] Knaster B., Kuratowski K.: Sur les ensembles connexes. Fund. Math. 2 (1921), no. 1, 206–256 (French). DOI 10.4064/fm-2-1-206-255
[17] Knopfmacher J., Porubský Š.: Topologies related to arithmetical properties of integral domains. Exposition Math. 15 (1997), no. 2, 131–148. MR 1458761
[18] Steen L. A., Seebach J. A., Jr.: Counterexamples in Topology. Dover Publications, Mineola, 1995. MR 1382863 | Zbl 0386.54001
[19] Stevenhagen P., Lenstra H. W., Jr.: Chebotarëv and his density theorem. Math. Intelligencer 18 (1996), no. 2, 26–37. DOI 10.1007/BF03027290 | MR 1395088
[20] Sury B.: Frobenius and his density theorem for primes. Resonance 8 (2003), no. 12, 33–41. DOI 10.1007/BF02839049
[21] Szczuka P.: The connectedness of arithmetic progressions in Furstenberg's, Golomb's, and Kirch's topologies. Demonstratio Math. 43 (2010), no. 4, 899–909. MR 2761648
[22] Szczuka P.: The Darboux property for polynomials in Golomb's and Kirch's topologies. Demonstratio Math. 46 (2013), no. 2, 429–435. MR 3098036
[23] Wang S.: A counter-example to Grunwald's theorem. Ann. of Math. (2) 49 (1948), 1008–1009. DOI 10.2307/1969410 | MR 0026992
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