§ Topological proof of infinitude of primes

We take the topological proof and try to view it from the topology as semidecidability perspective.

• Choose a basis for the topology as the basic open sets$S(a, b) = \{ a n + b : n \in \mathbb Z \} = \{ mathbb Z + b \}$.This set is indeed semi-decidable. Given a number $k$, I can check if$(k - b)$. So this is our basic decidability test.
• By definition, $\emptyset$ is open, and $\mathbb Z = S(1, 0)$. Thus it isa valid basis for the topology. Generate a topology from this. So we arecomposing machines that can check in parallel if for some $i$,$(k - b[i])$ for some index.
• The basis $S(a, b)$ is clopen, hence the theory is decidable.
• Every number other than the units $\{+1, -1\}$ is a multiple ofa prime.
• Hence, $\mathbb Z \setminus \{ -1, +1 \} = \cup_{p \text{prime}} S(p, 0)$.
• Since there a finite number of primes [for contradiction], the right hand sidemust be must be closed.
• The complement of $\mathbb Z \setminus \{ -1, +1 \}$ is $\{ -1, +1 \}$. Thisset cannot be open, because it cannot be written as the unionof sets of the form $\{ a n + b \}$: any such union would have infinitelymany elements. Hence, $\mathbb Z \setminus \{ -1, +1 \}$ cannot be closed.