If τ is the finite-closed topology, then the semi-open sets are precisely the open sets.
We can assume the set is infinite, otherwise the proof is trivial. I can easily show that open sets are semi-open. Any open set A are element of X without a finite number of points. If we remove another point, it is still an open set, O, and is a subset of A. The closure of O is the entire set because that is the smallest closed set that contains O since every closed set is finite. So every open set fits the criterion for being semi open. Now I need to show that if there are any other semi open sets in the topology, is is just part of the open sets. I am not sure how to show this.