What are the continuous functions $mathbb{N} to mathbb{R}$ when $mathbb{N}$ is given the coprime topology?

Question

Consider $mathbb{N}$ with the topology generated by sets of the form $U_{a,b}=mathbb{N}  cap  {an+b: n in mathbb{Z}} $ where $text{gcd}(a,b)=1$.
What are the continuous functions $mathbb{N} to mathbb{R}$ when $mathbb{N}$ is given this topology and $mathbb{R}$ has the Euclidean topology?
Some thoughts: if the range of such a continuous function were infinite, it would imply we could partition $mathbb{N}$ into countably infinite many disjoint sets of the form $U_{a,b}$ as above, which somehow seems unlikely.

Henno Brandsma · Accepted Answer

The space (see here) is connected and countable, so its continuous image in $Bbb R$ is also connected and countable hence a singleton. So all such maps are constant.

What are the continuous functions $mathbb{N} to mathbb{R}$ when $mathbb{N}$ is given the coprime topology?

One Answer

Add your own answers!

Ask a Question