  This paper provides a canonical construction of a Noetherian least fixed
point topology. While such least fixed point are not Noetherian in general, we
prove that under a mild assumption, one can use a topological minimal bad
sequence argument to prove that they are. We then apply this fixed point
theorem to rebuild known Noetherian topologies with a uniform proof.
  In the case of spaces that are defined inductively (such as finite words and
finite trees), we provide a uniform definition of a divisibility topology using
our fixed point theorem. We then prove that the divisibility topology is a
generalisation of the divisibility preorder introduced by Hasegawa in the case
of well-quasi-orders.
