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.