I 've been told that every single point of a topology T is closed if and only cofinite topology is contained in T. I am struggling to prove this. I was thinking that indeed if every single element is closed then every finite set is closed(because is a union of points which are closed).Which is good because then then the complement of the open sets corresponding to the closed sets is finite. But what i cannot prove is why can't we have an infinite closed set ?
You can have an infinite closed set! You have shown if every point is closed, then every cofinite set is open. In particular, every cofinite set is in $T$. However, you can have infinite closed sets without contradiction. To see this concretely, just look at $\mathbb R$ with the usual topology. Finite sets are closed, so cofinite sets are open. On the other hand, infinite sets like $[0,1]$ are also closed.