Is there a name for this type of topology?

Let X be an infinite set. Let A be an infinite subset of X.

Definite a topology, T, on X, where every open set is either:

1. The empty set, or

2. Contains all but finitely many points of A. (Alternatively put: to form opens of this type, take the set A, remove finitely many points, then add any amount of points from X. )

T contains the empty set, and X contains all but finitely many points from A, so X is in T. If O and U both contain all but finitely many points of A, then so will the intersection. And an arbitrary union of this type will also be in T.

If the difference between A and X is finite, then T is simply the co-finite topology. But is there a name for the topology when X minus A is infinite?

Thanks in advance.