Is the Opposite of the Open Closed in Topology? - On

Question

Following is the topological proof of "infinitude of primes"

If you see above proof, it first defines its own toplogy and comments

" This topology has two notable properties... 1. the complement of a finite set cannot be a closed set."

I understood the complement of a finite set on $\Bbb Z$ results in infinite set, thus it could be possibly become open if it meets its topology-definition. However, why it suddenly says "cannot be a closed set"?

It looks like I am confusing something. Help me to figure out where I wrongly misunderstanding.

Answer 1

What it says is that the complement of a finite (non-empty) set cannot be closed. This is so because, for this topology, a non-empty finite set cannot be open, since every non-empty open set contains an infinte sequence.

Answer 2

Start by a "a finite set $F$ cannot be open". This is true if $F \neq \emptyset$ (the empty set is finite and is explicitly mentioned as being open, as it should be in any topology), because if we have $x \in F$ we have a whole infinite sequence $x \in S(a,b) \subseteq F$, which cannot happen if $F$ is finite.

A set $C \subseteq \mathbb{Z}$ is closed by definition if $\mathbb{Z}\setminus C$ is open. So if $\mathbb{Z}\setminus C$ is finite and $\mathbb{Z} \neq C$ (i.e. the complement of $C$ is non-empty and finite), then the complement of $C$ cannot be open, and so $C$ cannot be closed.