Struggling to understand Furstenberg's proof of the infinitude of prime number.
It seems like a finite union of the closed arithmetic progressions $\bigcup_{b\in\mathbb{N}_a}{S(a,b)}$ would have to be closed as it seems to be a finite union of closed sets, however it's also the set of the integers $\mathbb{Z}$ which cannot be closed so there's a contradiction.
Is there some rule here that prohibits the finite union on the free term $b$ in the progression $a\mathbb{N}+b$? Is it only finite when we index $a\in\mathbb{N}$?
I'm also wondering about a finite union of progressions $\bigcup_{a\in A}{a\mathbb{Z}+f(a)}$ with $f(a)\in\mathbb{N}_a$ where $A$ is a finite set on natural numbers. Intuitively it also seems to be closed.
It's also intuitive that $2\mathbb{Z}\cup2\mathbb{Z}+1$ is supposed to be a finite union of two clopen sets which is supposed to be closed but isn't so there's a contradiction, is there something I misundertood about this topology?