Understanding a Topological proof of the infinity of primes

Question

Sorry if the question is too basic, I haven't taken any Topology lectures.

I found this proof in "Proofs from THE BOOK" and don't quite understand the explanation of why $ N_{a,b} $ is closed, I hope you could spell it out for me.

Thank you in advance.

Answer 1

In any topology, a union of open sets is open.

In this topology, each $N_{a+i,b}$ is open, so $\bigcup\limits_{i=1}^{b-1} N_{a+i,b}$ is open.

In any topology, the complement of an open set is closed.

Therefore, $N_{a,b}=\Bbb Z\setminus \bigcup\limits_{i=1}^{b-1} N_{a+i,b}$ is closed.

(In fact, $N_{a,b}$ is both open and closed -- called clopen -- in this topology.)