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 pro
ve 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.
