Show that $U$ is an open subset of a topological space $X$ if and only if for every point $z\in U$ each net converging to $z$ is co-finitely contained in $U$.
Is this question as stated correct? IMO, I think it's supposed to say eventually in $U$ rather than co-finitely contained in $U$.
No, it’s not. Let the index set be $\Bbb Z$ in the usual order and define the net $f:\Bbb Z \to \Bbb R$ be defined by $f(n)=\min(0,n)$. Then $f$ converges to $0$ but infinitely many terms of the net (for $n<0$) are outside the open neighbourhood $(-1,1)$ of $0$.
It’s indeed more customary and correct to say that the net is eventually in the open set. Tails need not be cofinite as it is the case for sequences.