The Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $a,b$ relatively prime.

It is not known if the Golomb space admits a non-trivial homeomorphisms. On the other hand, $\mathbb G$ has continuum many continuous self-maps. In particular, each polynomial $f:\mathbb G\to\mathbb G$, $f:x\mapsto a_1x+\dots+a_nx^n$, with integer coefficients and without the constant term is a continuous self-map of $\mathbb G$. But all known examples of non-constant continuous self-maps $f$ of the Golomb space have the property that for any $x\in\mathbb N$ each prime divisor $p$ of $x$ divides $f(x)$.

Question 1. Is there a non-constant continuous map $f:\mathbb G\to\mathbb G$ such that $f(x)=1$ for some $x\ne 1$?

We can also ask a more general

Question 2. Is it true that any map $f:F\to\mathbb G$ defined on a finite subset $F\subset\mathbb G$ extends to a continuous self-map $\bar f:\mathbb G\to\mathbb G$ of $\mathbb G$?

Remark. The last question is equivalent to asking if the set $C(\mathbb G)$ of all continuous self-maps of the Golomb space is dense in the space $\mathbb N^{\mathbb N}$ of all functions of $\mathbb N$, where $\mathbb N^{\mathbb N}$ is endowed with the Tychonoff product topology of discrete spaces $\mathbb N$.

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    $\begingroup$ I assume you mean "non-constant continuous map" in Question 1? $\endgroup$ – Greg Martin Nov 23 '17 at 8:17
  • $\begingroup$ @GregMartin Thank you for the comment. Indeed I assumed that $f$ is continuous. $\endgroup$ – Taras Banakh Nov 23 '17 at 8:41
  • $\begingroup$ @TarasBanakh Out of pure curiosity, what's the motivation and/or ultimate desire with knowing this particular info about the Golomb space (since you and someone else asked a few questions about it recently)? $\endgroup$ – Chris Gerig Nov 25 '17 at 2:00
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    $\begingroup$ @ChrisGerig This is a long story. I will try to explain it in several comments (one is too short). The interest to the Golomb space (which was initially introduced by Brown) was warmed up by Solomon Golomb (known popularizator of mathematics). Golomb (who made his Ph.D. studying the distribution of primes) noticed that the famous Dirichlet Theorem is equivalent to the density of prime numbers in the Golomb topology, and made a conjecture that this non-elementary theorem can be proved by topological tools (like the infinite of primes proved topologically by Furstenberg). $\endgroup$ – Taras Banakh Nov 25 '17 at 8:39
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    $\begingroup$ @ChrisGerig This motivates the interest to the Golomb space. Besides this, this space is quite exotic: it is countable, connected and Hausdorff, probably it is the simplest example of such space. On the other hand, simple questions on the structure of this space turns out to be hard. For example, we do not know if this space admit non-trivial homeomorphisms. All known non-constant continuous self-maps of this space have property that all prime divisors of any number x divide also its image f(x). So, the question actually asks about the existence of continuous maps wiuthout this property. $\endgroup$ – Taras Banakh Nov 25 '17 at 8:44

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