  We introduce two families of transcendental numbers which we call finite
factorial (FF) and partially finite factorial (PFF) numbers respectively, with
the former one being subfamily of the latter one. These numbers arise naturally
from some transcendental criterion for real numbers via their $b$-ary
expansions. We show that rational numbers (eventually periodic words) can not
be finite factorial. Then we consider the geometric (topological) properties of
the collection of all the FF numbers, including its countability, density and
Hausdorff dimension. Some numerical examples are given to illustrate certain
results in the work.
