On continuous polynomials of the Macías space
Keywords:
Polynomials, exponential functions, Golomb’s topologyAbstract
To my wife and our first child
Let $\mathbb{N}$ be the set of natural numbers. The Macías space $M(\mathbb{N})$ is the topological space $(\mathbb{N}, \tau_M)$ where $\tau_M$ is generated by the collection of sets $\sigma_n:=\{m \in \mathbb{N}: \operatorname{gcd}(n, m)=1\}$. In this paper, we characterize the continuity of polynomial and exponential functions over $M(\mathbb{N})$; and prove that the only continuous polynomials are monomials, and that the only continuous exponential functions are of the form $f(x)=a^x$.
References
T. M. Apostol, Introduction to Analytic Number Theory, Springer, 2013.
T. Banakh, J. Mioduszewski, and S. Turek, On continuous self-maps and homeomorphisms of the Golomb space, Comment. Math. Univ. Carolin. 59 (2018), no. 4, 423-442.
T. Banakh, D. Spirito, and S. Turek, The Golomb space is topologically rigid, Comment. Math. Univ. Carolin. 62 (2021), no. 3, 347-360.
M. Brown, A countable connected Hausdorff space, Bull. Amer. Math. Soc. 59 (1953), 367.
P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Abh. Königl. Preuß. Akad. Wiss.
$45(1837), 81$.
H. Furstenberg, On the infinitude of primes, Amer. Math. Monthly 62 (1955), no. 5,353 .
S. W. Golomb, A connected topology for the integers, Amer. Math. Monthly 66 (1959), no. 8, 663-665.
J. Macías, On self-homeomorphisms of the Macías space, Topology Appl. 373 (2025), 109520.
J. Macías, Another Topological Proof of the Infinitude of Prime Numbers, Integers 24 (2024), no. 47.
J. Macías, The Macías topology on integral domains, Topology Appl. 357 (2024), 109070.
R. Murty and N. Thain, Primes in certain arithmetic progressions, Funct. Approx. Comment. Math. 35 (2006), 249-259.
I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, John Wiley & Sons, 1991.
I. Schur, Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, 1912.
L. A. Steen and J. A. Seebach, Counterexamples in Topology, Springer-Verlag, 1978.
P. Szczuka, The Darboux property for polynomials in Golomb's and Kirch's topologies, Demonstratio Math. 46 (2013), no. 2, 429-435.
