Topology of naturals For $a, r \in \mathbb{N}$, be $B_{a,r} = \{a + kr : k\in \mathbb{N}\}$.
(a) Show that $B = \{B_{a,r} : a, r \in \mathbb{N}, r\neq 0, < r\}$ is a basis for a
topology on $\mathbb{N}$.
(b) Show that in this topology, each $B_{a,r}$ is a closed one.
(c) Write $\mathbb{N}-\{1\}$ as a closed meeting.
(d) Conclude that there are infinite primes.

*

*First who are the $\mathbb{N}$ topology open?

*To prove that the $B_{a,r}$ collection is a basis for the $\mathbb{N}$ topology should I show that joining base elements generates any element sofa in the topology?

*Does it make sense $B_{a,r}$ to be closed, because it is elements that joined generate the elements of the topology that are opened?

*Why proving the letter (a),(b) and (c) can at the end conclude that there are infinite prime numbers? I didn't understand anything.

 A: I'm gonna leave (a) and (b) to you because I think you should be able to prove such things by yourself (also I'm too lazy to do it). Just to facilitate (d), here's the answer for (c):
$\mathbb{N} \setminus \{1\} = \cup \{B_{0,p} : p$ is a prime $\}$. This is because every natural number that is not $1$ is a product of primes.
By (b), we know also that each $B_{0,p}$ is closed.
Suppose $A \subseteq \mathbb{N}$ is open. Then by (a), as the $B_{a,r}$ form a basis, there must be some $B_{a,r}$ such that $B_{a,r} \subseteq A$ and thus $A$ is infinite (clearly the $B_{a,r}$ are infinite). This tells us that finite sets can't be open. Now we can put all the pieces together:
Suppose there were finitely many primes. Then $\cup \{B_{0,p} : p$ is a prime $\}$ is a finite union of closed sets, and thus closed- meaning $\mathbb{N} \setminus \{1\}$ is closed. Thus the complement is open- $\{1\}$ is open. But we just estabilished that finite sets can't be open. Thus we have reached a contradiction.
A: By definition, the open sets of your topology will be (possibly infinite) unions of elements $B_{a,r}$ of your given basis.
Yes, it makes sense for each $B_{a,r}$ to be also closed (such a subspace is sometimes called a clopen set, as it's both closed and open). Notice though that this does not imply that every open set is also closed, because infinite unions of closed sets are not necessarily closed.
The exercise is basically an excuse to get to the final point, which is known as the
Furstenberg's proof of the infinitude of primes. Basically, with such a topology on the set of natural numbers, any (non-empty) open set contains infinite elements, so no finite set can be open. If you assume that prime numbers are finite, then the union of all (closed sets) $B_{p,0}$ such that $p$ is prime would still be closed. But, as you'll find at c),
$$ \bigcup_{p\text{ prime}} B_{p,0}=\mathbb N\smallsetminus\{1\},$$
which can't be closed because the finite set $\{1\}$ cannot be open. I'll leave you to work out the details!
