Communities

Writing
Writing
Codidact Meta
Codidact Meta
The Great Outdoors
The Great Outdoors
Photography & Video
Photography & Video
Scientific Speculation
Scientific Speculation
Cooking
Cooking
Electrical Engineering
Electrical Engineering
Judaism
Judaism
Languages & Linguistics
Languages & Linguistics
Software Development
Software Development
Mathematics
Mathematics
Christianity
Christianity
Code Golf
Code Golf
Music
Music
Physics
Physics

Dashboard
Notifications
Mark all as read
Q&A

Can we adapt Furstenberg's proof of the infinitude of primes to other interesting subsets of the integers?

+5
−0

In a textbook, I was introduced to Furstenberg's topological proof of the infinitude of primes, which goes basically as follows:

We define a topology on $\mathbb{Z}$ such that the sets $$S(a,b)=\set{an+b|n\in\mathbb{N}}$$ for $a$ and $b$ integers, with $a\neq0$, as well as the empty set, are open. You can show without too much work that these sets are also closed. Since the only numbers that are not multiples of some prime are $1$ and $-1$, we can write $$\mathbb{Z}\setminus\set{-1,1}=\bigcup_{p\text{ prime}}S(p,0)\tag{1}$$ $\set{-1,1}$ is not open, and $\mathbb{Z}\setminus\set{-1,1}$ is not closed. Therefore, the right-hand side must be the union of an infinite number of closed sets, which implies an infinite number of primes.

Now, this isn't truly a "topological" proof - it uses no theorems from topology. In fact, we can show that Furstenberg's proof is quite similar to Euclid's classic proof of the infinitude of primes, just somewhat couched in topological language. All the same, I found it helpful to think of the argument in Furstenberg's terms, which led me to a question:

Can we define a different topology on $\mathbb{Z}$ from which we can directly prove the infinitude of another interesting, non-trivial subset of the integers - perhaps a particular type of prime? I would guess maybe not, as the left-hand side of our new version of $(1)$ might end up being of the form $\mathbb{Z}\setminus\set{\text{some open set}}$, and so the proof would break down - but I don't really have anything to support this, and I'm not well-enough versed in topology to have confidence in it.

Why does this post require moderator attention?
You might want to add some details to your flag.
Why should this post be closed?

2 comments

I'm not so well-versed in the relevant maths, but it would seem to be that the inverse would hold — any set containing the infinite set of primes must itself be infinite, though not necessarily of the same cardinality. DonielF‭ 4 months ago

@DonielF That's quite true, though I'm less interested in sets containing all of the primes and more interested in subsets of primes. HDE 226868‭ 4 months ago

0 answers

Sign up to answer this question »

This community is part of the Codidact network. We have other communities too — take a look!

You can also join us in chat!

Want to advertise this community? Use our templates!