Abstract
If X is a set, τ is not a discrete topology on X then τ is called an extremal topology if every topology which is strictly finer than τ is discrete. The main purpose of this paper is to prove an existence theorem for extremal topologies and to prove a second theorem, which determines how an extremal topology on a finite set looks. By using these two theorems we prove a counting theorem which gives the number of extremal topologies on a set with n elements.